March 03, 2025
We prove a necessary criterion for the (non-)existence of nontrivial solutions to the Dirac equation \(D\psi=A\cdot_{\text{Cl}}\psi\) on Riemannian manifolds that are either closed or of bounded geometry. This generalizes a result of [1] on \(\mathbb{R}^n\) where the criterion relates the \(L^n\)-norm of \(A\) to the Sobolev constant on \(\mathbb{R}^n\). On Riemannian manifolds the role of the Sobolev constant will be replaced by the Yamabe invariant. If \(n\) is odd, we show that our criterion is sharp on \(\mathbb{S}^n\).
In [1] R. Frank and M. Loss prove a sharp necessary condition for the (non-)existence of nontrivial solutions to the Dirac equation
\[\begin{align} \gamma (-i \nabla - A)\psi =0 \text{ in } \mathbb{R}^n \text{ for } n\geq 3. \label{eq:Rnmaineq}
\end{align}\tag{1}\] Solutions to this equation are called zero modes and were first derived in \(3\) dimensions in [2]. They arise in several areas of mathematics and are particularly related to the size of magnetic fields in physics. We refer the reader to [3] and the references therein for more background information and material. In the setting in [1] the quantity \(A\) is a vector field on \(\mathbb{R}^n\) and \(\psi\) is a spinor, which in this context is a \(\mathbb{C}^{2^{[n/2]}}\) valued
function. In the context of 1 , the term \(\gamma A\psi\) stands for the Clifford action of \(A\) on \(\psi\) and \(\gamma (-i\nabla)\psi\) stands for the Dirac operator on \(\mathbb{R}^n\).
In this paper, we will attack the question of whether a similar statement holds on Riemannian manifolds and its associated spinor bundles. For a vector field \(X\) on \(M\), we denote by
\(X\cdot_{\text{Cl}}\psi\) the Clifford action of \(X\) on a spinor \(\psi\). Note that for \(p\in M\), in contrast to the
convention in [1], in local coordinates the Clifford algebra is determined by the multiplication rule \[\begin{align} e_i
e_j+ e_j e_i=-2\delta_{ i j}
\end{align}\] for an orthonormal basis \((e_i)_{i=1,...,n}\) of \(T_pM\). By translating 1 into the setting of Riemannian manifolds, we investigate the
existence of nontrivial solutions to \[\begin{align} A \cdot_{\text{Cl}}\psi = D\psi \label{eq:maineq} \text{ on } M.
\end{align}\tag{2}\] Throughout, we assume that \((M,g)\) is a complete, smooth Riemannian manifold without boundary and \(\dim M=n \geq3\) together with a Clifford bundle
\(\Sigma\), cf. Section 2.1 for precise definitions. Sections of the Clifford bundle are called spinors and will be mostly denoted by \(\psi\). The
associated Dirac operator will be denoted by \(D\) as long as the metric \(g\) is fixed. Note, that whenever we assume \(M\) to be spin we assume the spin
structure \(\sigma\) to be fixed and \(\Sigma\) will denote the associated spinor bundle to the \(\text{Spin}(n)-\)principal bundle over \((M,g,\sigma)\). Additionally we always assume \[\begin{align} A\in L^{n}(M,TM).
\end{align}\] The proof of [1] relies on the Euclidean Sobolev inequality, cf. [4], in \(\mathbb{R}^n\) given by \[\begin{align} S_n\big(\int_{\mathbb{R}^n} |u|^{\frac{2n}{n-2}}dx\big)^{\frac{n-2}{n}} \leq \int_{\mathbb{R}^n} |\nabla u |^2dx
\quad\text{for } u \in H^{1,2}(\mathbb{R}^n).
\end{align}\] Here, \(S_n\) denotes the optimal constant in the Sobolev inequality, which was first independently derived in [5] and [6]. Unfortunately this inequality is no longer true on general manifolds. However, by [7] there is a close relation of the optimal constant in the Sobolev inequality on \(\mathbb{R}^n\) and the so called Yamabe invariant on \(\mathbb{S}^n\). This suggests to employ an inequality, based on the Yamabe invariant, cf. Section 2. The Yamabe invariant, denoted by \(Y(M,[g])\), is a
conformal invariant, i.e., for a Riemannian metric \(g\) it is independent on the choice of a metric belonging to the conformal class of \(g\).
It plays an important role in the solution of the original Yamabe problem, which affirmatively answers the question whether there always exists a metric of constant scalar curvature that is conformally equivalent to a given metric on a closed Riemannian
manifold \((M,g)\). For more details on the original Yamabe problem we refer to [8]–[12] and to [13] and [14] for some results
on non-compact manifolds.
Before stating our main results let us finally mention that the Bär-Hijazi-Lott invariant relates problems involving the Yamabe invariant and Dirac operators through the Hijazi inequality, cf. [15]. Even if this may not be directly related to the results in the present paper, it may be interesting to keep this in mind for future work to see whether it is possible to connect our results somehow with spinorial
Yamabe type problems.
Our first main result provides a necessary condition for nontrivial solutions to 2 on closed manifolds under similar conditions as in [1].
In our second main result, we prove a similar statement for a class of non-compact manifolds, namely manifolds of bounded geometry. As some regularity issues appear we impose stronger initial regularity conditions on the solution compared to Theorem [thm:spin].
We introduce notations and collect geometric and analytic preliminaries in Section 2. In Section 3, we state and prove an integral equality, which serves as the main starting point for the proof of Theorem [thm:spin]. Section 4 is devoted to proving of a slightly more general version of Theorem [thm:spin]. Here we also establish a similar theorem for manifolds of bounded geometry. In the last section we recover the case of equality on the sphere and characterize manifolds on which equality is possibly attained in [eq:inequYamspin].
I would like to thank my supervisor Nadine Große for many important and enlightening discussions and suggestions about Dirac operators and spinorial problems in general. I also want to thank my second supervisor Guofang Wang for further suggestions and for bringing attention to the problem. Finally, I want to thank Rupert Frank for a very useful discussion during his visit at the University of Freiburg.
Let \((M,g)\) be a smooth and complete Riemannian manifold with \(\dim M=n\). Let \((\Sigma \rightarrow M,\langle \cdot, \cdot \rangle, \nabla^{\Sigma})\)
be a bundle of Clifford modules over \(M\) equipped with a hermitian metric \(\langle \cdot,\cdot \rangle\) and a metric connection \(\nabla^\Sigma\). If the
Clifford multiplication is skew-adjoint with respect to \(\langle\cdot,\cdot \rangle\) and the connection \(\nabla^{\Sigma}\) is compatible with the Levi-Civita connection on \(M\) and the Clifford multiplication, we call \(\Sigma\) a Clifford bundle over \(M\), cf. [16]. Sections of \(\Sigma\) will be called spinors. We define the Dirac operator \(D_g\) associated with \(\Sigma\)
by the composition of the connection \(\nabla^{\Sigma}\) and the Clifford multiplication, cf. [16]. In terms of a local orthonormal
frame \(D_g\) is given by \[\begin{align} D_g \psi = \sum^n_{i=1}e_i \cdot_{\text{Cl}}\nabla_{e_i}^{\Sigma} \psi, \label{eq:dirlocal}
\end{align}\tag{3}\] where \(\psi: M \rightarrow \Sigma\). If we are working with a fixed metric \(g\), we will write \(D=D_g\). An important
property of the Dirac operator is its close relation to the curvature of \(\nabla^{\Sigma}\). Following [16] there is an operator
\(\mathcal{K} \in \text{End}(\Sigma)\) satisfying the Weitzenböck formula \(D^2= (\nabla^{ \Sigma } )^{*}\nabla^{\Sigma} + \mathcal{K}\), where \(\mathcal{K}\) is locally given by \[\begin{align} \sum^n_{i<j} \left(e_j e_i\right) \cdot_{\text{Cl}}\left(\nabla_{e_j}\nabla_{e_i}-
\nabla_{e_i}\nabla_{e_j}\right) \label{eq:cliffconloc}
\end{align}\tag{4}\] for a synchronous orthonormal framing \((e_i)_{i=1,...,n}\), cf [16]. Note that at the origin of
a local orthonormal synchronous frame \(\nabla_{e_j}e_i=0\) and the Lie bracket of \(e_i\) and \(e_j\) vanishes for all \(i,j\). The operator \(\mathcal{K}\) is called the Clifford contraction. By [16], we have the
explicit formula \(\mathcal{K}=F^{^\Sigma} + \frac{1}{4}\text{scal}_g\), where \(\text{scal}_g\) denotes the scalar curvature of \((M,g)\) and the term \(F^{\Sigma}\) is the Clifford contraction of the curvature of \(\nabla^{\Sigma}\) and is known as the twisting curvature of \(\Sigma\).
If \(M\) admits a spin structure \(\sigma\), then \(\Sigma(M,g,\sigma)\) is called the spinor bundle over \(M\) associated
to the spin structure \(\sigma\). Note that whenever we talk about spinor bundles we always assume that \(\sigma\) is fixed. If \(g\) is fixed as well, we
will simply write \(\Sigma(M,g,\sigma)=\Sigma\).
We will now introduce the twistor operator \(T_g\) as it will be of great importance for the proof of the main theorem and the characterization of cases of equality in [eq:inequYamspin].
Lastly we introduce the notion of bounded geometry, which will be the assumption for the main theorem in the setting of non-compact manifolds.
Let \((M,g)\) be a smooth and complete Riemannian manifold, either compact or of bounded geometry. By \(\Gamma_{c}^{\infty}(M,\Sigma)\), we denote the smooth and compactly supported
spinors of \(\Sigma\). For \(\psi \in \Gamma^{\infty}_c(M,\Sigma)\) we define, for \(1\leq p<\infty\) the \(L^{p}\)-norm
of \(\psi\) by \[\begin{align} \|\psi\|_{L^p(M,\Sigma)}:=\left(\int_M |\psi|^p dx\right)^{\frac{1}{p}},
\end{align}\] where \(|\cdot|\) is understood as the fiber-wise norm induced by the hermitian metric and \(dx\) denotes the Riemannian volume element. Then for \(k \in \mathbb{N}_{0}\) we define \[\begin{align} \| \psi \|^p_{H^{k,p}} := \sum^k_{j=1} \|\nabla^{j} \psi \|^p_{L^p(M,\Sigma \otimes T^{*}M^{\otimes j })},
\end{align}\] as the Sobolev norm of \(\psi\). We define the Sobolev space \(H^{k,p}(M,\Sigma)\) as the closure of \(\Gamma^{\infty}_{c}(M,\Sigma)\)
with respect to \(\|\cdot \|_{H^{k,p}}\). Note, that there are many equivalent ways of defining Sobolev spaces on manifolds and vector bundles. Without any attempt of completeness [17], [18], [4],[16] provide good references for more details on the definition of Sobolev spaces on manifolds or vector bundles.
As the proof of [eq:gard] is reduced to a local coordinate patch by the means a partition of unity, the result still holds on manifolds of bounded geometry, cf. [19].
For a Riemannian manifold the Yamabe invariant is defined as \[\begin{align} Y(M,[g])= \inf_{\underset{v\neq0}{v \in C_c^\infty(M)}} \frac{\int_{M} vL_g v dvol_g}{\|v\|^2_{L^{\frac{2n}{n-2}}}},
\end{align}\] where the operator \(L_g= \frac{4(n-1)}{n-2} \Delta_g + \text{scal}_g\) is known as the conformal Laplacian. It is a conformal invariant, cf. [20], and is one of the main objects studied in Yamabe-type problems. Following [13], we remark that since we
take the infimum over compactly supported functions, we can use the same definition of the Yamabe invariant for non-compact manifolds.
Analogously to the proof of [1], the proof of Theorem [thm:spin] is based on an integrated version of the Schrödinger-Lichnerowicz formula, which will be established in this section. Throughout this section, let \((M,g)\) be a Riemannian manifold and \(\Sigma\) a Clifford bundle over \(M\), \(\dim M= n\geq3\). Similarly as in [1], to avoid problems concerning regularity and dividing by zero we consider the following regularization for \(\psi \in \Gamma(M,\Sigma)\): \[\begin{align} |\psi|_{\varepsilon}:= \sqrt{|\psi|^2+\varepsilon^2}, \qquad \varepsilon>0. \end{align}\] We will now state and prove the integral identity analogue to [1] and highlight the main differences to the flat case, where \(\psi: \mathbb{R}^n\rightarrow\mathbb{C}^N\). By \(\mathcal{K}\) we denote the Clifford contraction of the curvature of \(\Sigma\). We always assume that \(\mathcal{K}\) is a bounded section in the endomorphism bundle of \(\Sigma\). The proof follows the idea and notions of [1].
Proof. As in the proof of [1], the proof splits into 5 steps. We first note that by standard computations we have
\[\begin{align} \text{Re}\left(\left\langle \psi, \nabla^{\Sigma}\psi \right\rangle \right) =\left| \psi \right| \nabla^M \left|\psi\right| = |\psi|_{\varepsilon}\nabla^M |\psi|_{\varepsilon}.
\label{eq:step0}
\end{align}\tag{5}\] Note that \(\langle \psi, \nabla^{\Sigma}\psi \rangle\) is not the hermitian product of two spinors as \(\nabla^{\Sigma} \psi \in \Gamma(M,\Sigma\otimes T^*
M)\). It should be considered as a \(1\)-form with the local expression \(\langle \psi, \nabla^{\Sigma} \psi \rangle= \sum^n_{j=1} \langle \psi, \nabla^{\Sigma}_{e_j} \psi \rangle
e^b_j\). Here \(e_j^b\) denotes the musical isomorphism of \(T_pM\) and \(T^{*}_pM\) given by \(X^b(Y):=g(X,Y)\) for
\(X,Y\in T_pM\).
We use [eq:twistor95abs] for the integrand on the left hand side and observe \[\begin{align} \left|T_g
\big(\frac{\psi}{|\psi|_{\varepsilon}^{n/n-1}}\big)\right|^2 |\psi|_{\varepsilon}^2= \left(\left| \nabla^{\Sigma} \left(\frac{\psi}{|\psi|_{\varepsilon}^{n/n-1}}\right)\right|^2-\frac{1}{n}\left|D\left(\frac{\psi}{|\psi|_{\varepsilon}^{n/n-1}}\right)
\right|^2\right) |\psi|_{\varepsilon}^2.
\end{align}\] Then, using 5 , step (1) and (2) in [1] are just pointwise identities and the proof for sections in
\(\Gamma(M,\Sigma)\) can be adapted straightforward from the proof for sections in \(\Gamma(\mathbb{R}^n,\mathbb{C}^N)\). We obtain the identities \[\begin{align} \left|\nabla^{\Sigma} \left( \frac{\psi}{|\psi|_{\varepsilon}^{\frac{n}{n-1}}}\right)\right|^2 |\psi|_{\varepsilon}^2 = \frac{\left|\nabla^{\Sigma} \psi\right|^2}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}} + \left| \nabla^M
\left( |\psi|_{\varepsilon}^{\frac{n-2}{n-1}} \right) \right|^2\left[ \left(\frac{n}{n-2}\right)^2\frac{\left|\psi\right|^2}{|\psi|_{\varepsilon}^2} - \frac{2n(n-1)}{(n-2)^2} \right] \label{eq:step1}
\end{align}\tag{6}\] and \[\begin{align} \begin{aligned} \left|D\left(
\frac{\psi}{|\psi|_{\varepsilon}^{\frac{n}{n-1}}}\right)\right|^2|\psi|_{\varepsilon}^2=&\frac{\left|D\psi\right|^2}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}} + \left(\frac{n}{n-2}\right)^2\left|\nabla^M
\left(|\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\right)\right|^2\frac{\left|\psi\right|^2}{|\psi|_{\varepsilon}^2} \\ -&\frac{2n}{n-1}\frac{1}{|\psi|_{\varepsilon}^{\frac{2}{n-1}+1}}\text{Re}\left( \left\langle D\psi, \nabla^M
|\psi|_{\varepsilon}\cdot_{\text{Cl}}\psi \right\rangle \right). \end{aligned} \label{eq:step2}
\end{align}\tag{7}\] For step (3) let \(\chi \in C^{\infty}_{c}(M)\) such that \(\chi\) is supported in a small enough coordinate patch \(U\subset
M\), which admits a synchronous orthonormal frame \((e_i)_{i=1,...,n}\). We claim \[\begin{align} \begin{aligned} \int_{M}
|\psi|_{\varepsilon}^{-\frac{2}{n-1}} \left|\nabla^{\Sigma} \psi\right|^2 \chi dx &= \frac{2(n-1)}{(n-2)^2} \int_{M} \left| \nabla^M \left( |\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\right) \right|^2 \chi dx \\ +& \int_{M}
|\psi|_{\varepsilon}^{-\frac{2}{n-1}}\left|D \psi\right|^2 \chi dx \\ -& \frac{2}{n-1} \int_{M} |\psi|_{\varepsilon}^{-1-\frac{2}{n-1}} \text{Re} \left(\left\langle \left(\nabla^M |\psi|_{\varepsilon}\right) \cdot_{\text{Cl}}\psi, D\psi
\right\rangle\right) \chi dx \\ +&\sum_{\underset{j\neq k}{j,k=1}}^{n} |\psi|_{\varepsilon}^{-\frac{2}{n-1}}\text{Re}\left(\left\langle e_j \cdot_{\text{Cl}}\psi, e_k \cdot_{\text{Cl}}\nabla^{\Sigma}_{e_k} \psi\right\rangle\right) \nabla^M_{e_j} \chi
dx \\ -& \int_{M} |\psi|_{\varepsilon}^{-\frac{2}{n-1}}\text{Re}\left(\left\langle \psi, \mathcal{K} \psi \right\rangle\right) \chi dx. \end{aligned} \label{eq:step3}
\end{align}\tag{8}\] The proof of 8 is based on the identity \[\begin{align} \begin{aligned} \sum^n_{k< j}&\int_{M} f \left\langle
\nabla^\Sigma_{e_k} \psi, e_k e_j\cdot_{\text{Cl}}\nabla^\Sigma_{e_j} \psi \right\rangle dx+\int_{M} f \left\langle \nabla^\Sigma_{e_j} \psi, e_j e_k\cdot_{\text{Cl}}\nabla^\Sigma_{e_k} \psi \right\rangle dx\\=& -\sum^n_{k< j }\int_M\left(
\left(\nabla^M_{e_k}f\right) \left\langle \psi, e_k e_j \cdot_{\text{Cl}}\nabla^\Sigma_{e_j} \psi \right\rangle + \left(\nabla^M_{e_j}f\right) \left\langle \psi, e_j e_k \cdot_{\text{Cl}}\nabla^\Sigma_{e_k} \psi \right\rangle \right)dx\\ &- \int_{M} f
\left\langle\psi,\mathcal{K}\psi\right\rangle dx. \end{aligned} \label{eq:step3integralident}
\end{align}\tag{9}\] for \(f=|\psi|_{\varepsilon}^{-2/(n-1)}\chi\). Before we prove 9 , we make sure that every integral above is well defined.
As \(f\) and is bounded and \(\psi \in L^2(M,\Sigma)\), \(\nabla\psi \in L^2(M,\Sigma \otimes T^*M)\) the left hand side and the last integral on the right
hand side are well defined. For the first integral on the right hand side we note \[\begin{align} \nabla^M_{e_j} f = -\frac{2}{n-1} |\psi|_{\varepsilon}^{-\frac{2}{n-1}-1}
\left(\nabla^M_{e_j} |\psi|_{\varepsilon}\right)\chi + |\psi|_{\varepsilon}^{-\frac{2}{n-1}}\left(\nabla^M_{e_j}\chi\right). \label{eq:step3help3}
\end{align}\tag{10}\] Again, as \(\psi \in L^2(M,\Sigma)\), \(\nabla \psi \in L^2(M,\Sigma\otimes T^*M)\) and \(|\psi|_{\varepsilon}^{-2/(n-1)-1}\nabla_{e_j} \chi\) is bounded the integral involving \(\nabla_{e_j} \chi\) is well defined. For the integral involving \(\nabla^M_{e_j}|\psi|_{\varepsilon}\) we observe \[\begin{align} \begin{aligned} &\sum_{\underset{j\neq k }{j,k=1}}^{n}
\left(\nabla^M_{e_j}|\psi|_{\varepsilon}\right) \left\langle e_j \cdot_{\text{Cl}}\psi, e_k \cdot_{\text{Cl}}\nabla^{\Sigma}_{e_k} \psi \right\rangle\\ =& \sum_{j,k=1}^n \left(\nabla^M_{e_j}|\psi|_{\varepsilon}\right) \left\langle e_j
\cdot_{\text{Cl}}\psi, e_k \cdot_{\text{Cl}}\nabla^{\Sigma}_{e_k} \psi \right\rangle - \sum_{j=1}^{n} \nabla^M_{e_j} |\psi|_{\varepsilon}\left\langle \psi, \nabla^{\Sigma}_{e_j} \psi\right\rangle \\ =& \left\langle \nabla^M
|\psi|_{\varepsilon}\cdot_{\text{Cl}}\psi, D\psi \right\rangle - g\left(\nabla^M |\psi|_{\varepsilon},\left\langle \psi, \nabla^{\Sigma} \psi \right\rangle \right). \label{eq:step3help3local} \end{aligned}
\end{align}\tag{11}\] Hence, the diamagnetic inequality \(|\nabla^{\Sigma}\psi| \leq |\nabla^M |\psi||\), \(|\psi|/|\psi|_{\varepsilon}\leq 1\) and \(|\psi|_{\varepsilon}\nabla^M |\psi|_{\varepsilon}= |\psi| \nabla^M |\psi|\) imply \[\begin{align} &\int_M|\psi|_{\varepsilon}^{-\frac{2}{n-1}-1} \left| \sum_{\underset{j\neq k }{j,k=1}}^{n}
\left(\nabla^M_{e_j}|\psi|_{\varepsilon}\right) \left\langle e_j \cdot_{\text{Cl}}\psi, e_k \cdot_{\text{Cl}}\nabla^{\Sigma}_{e_k} \psi \right\rangle \right| \chi dx\\ =&\int_M|\psi|_{\varepsilon}^{-\frac{2}{n-1}-1}\left[ \left\langle \nabla^M
|\psi|_{\varepsilon}\cdot_{\text{Cl}}\psi, D\psi \right\rangle - g\left(\nabla^M |\psi|_{\varepsilon},\left\langle \psi, \nabla^{\Sigma} \psi \right\rangle \right) \right] \chi dx\\ \leq& C_{\chi,\varepsilon,n}\int_M
|\psi|_{\varepsilon}^{-\frac{2}{n-1}} \left[ \left|\nabla^{\Sigma} \psi \right| \left| D\psi\right|+\left|\nabla^{\Sigma}\psi \right|^2\right] dx\\ \leq&\tilde{C}_{\chi,\varepsilon,n}\int_M \left|\nabla^{\Sigma} \psi\right|^2 dx <\infty.
\end{align}\] Thus, every integral in 9 is well defined.
By density of \(C^{\infty}_{c}(M,\Sigma)\) in \(H^1(M,\Sigma)\), we prove 9 for \(\psi \in
C^{\infty}_{c}(M,\Sigma)\). To see 9 , we observe that \(h_{k}:= f\langle \psi, e_k e_j \cdot_{\text{Cl}}\nabla^{\Sigma}_{e_j}\psi\rangle\) is compactly supported on \(U\subset M\). For the vector field \(Y=h_k e_k\) on \(M\) we have \(\text{div}(Y)dx=\nabla^M_{e_k}h_kdx\). Thus, the divergence
theorem implies for any \(k\) \[\begin{align} 0=\int_M \nabla^M_{e_k}h_kdx. \label{eq:divergencethm}
\end{align}\tag{12}\] Hence, 12 and metric compatibility of \(\langle \cdot, \cdot \rangle\) and \(\nabla^{\Sigma}\) yield
\[\begin{align} \begin{aligned} &\int_{M} f \left\langle \nabla^\Sigma_{e_k} \psi, e_k e_j\cdot_{\text{Cl}}\nabla^\Sigma_{e_j} \psi \right\rangle dx+\int_{M} f \left\langle
\nabla^\Sigma_{e_j} \psi, e_j e_k\cdot_{\text{Cl}}\nabla^\Sigma_{e_k} \psi \right\rangle dx\\=& -\int_M\left( \left(\nabla^M_{e_k}f\right) \left\langle \psi, e_k e_j \cdot_{\text{Cl}}\nabla^\Sigma_{e_j} \psi \right\rangle + \left(\nabla^M_{e_j}f\right)
\left\langle \psi, e_j e_k \cdot_{\text{Cl}}\nabla^\Sigma_{e_k} \psi \right\rangle \right)dx\\ &- \int_{M} \left( f \left\langle \psi, \nabla^\Sigma_{e_k}\left(e_ke_j \cdot_{\text{Cl}}\nabla^\Sigma_{e_j}\psi\right)\right\rangle dx + f \left\langle
\psi, \nabla^\Sigma_{e_j}\left(e_je_k \cdot_{\text{Cl}}\nabla^\Sigma_{e_k}\psi\right)\right\rangle\right) dx. \end{aligned} \label{eq:step3help1}
\end{align}\tag{13}\] For the second integral let \(p \in M\) be the origin of the synchronous framing then \[\begin{align} \nabla_{e_j}e_k \big|_{p}=0
\end{align}\] for all \(j\) and \(k\). Summing in 13 over \(k<j\), the last integral yields the pointwise equation
\[\begin{align} \begin{aligned} f\left\langle \psi, \sum^n_{k<j} \left(\nabla^\Sigma_{e_k} \left(e_k e_j \nabla^\Sigma_{e_j} \psi\right) - \nabla^\Sigma_{e_j}\left(e_k e_j
\nabla^\Sigma_{e_k} \psi \right)\right)\right\rangle\big|_p =f\left\langle\psi, \mathcal{K}\psi \right\rangle\big|_p, \end{aligned} \label{eq:step3help2}
\end{align}\tag{14}\] where the right hand side is independent on the choice of the synchronous frame, cf. [16]. Summing over \(k<j\) in 13 and using 14 we conclude 9 . Moreover, as \(\chi\) is supported in a nice
coordinate patch we can integrate over the local expression \[\begin{align} \begin{aligned} &\left|\nabla^\Sigma \psi\right|^2- \left|D\psi\right|^2 \\ =&- \sum^n_{k<j}\left(
\left\langle e_j \cdot_{\text{Cl}}\nabla^\Sigma_{e_j} \psi, e_k \cdot_{\text{Cl}}\nabla^\Sigma_{e_k} \psi \right\rangle+ \left\langle e_k \cdot_{\text{Cl}}\nabla^\Sigma_{e_k} \psi, e_j \cdot_{\text{Cl}}\nabla^\Sigma_{e_j} \psi \right\rangle \right),
\end{aligned} \label{eq:step3help4}
\end{align}\tag{15}\] insert 9 and obtain \[\begin{align} \begin{aligned} &\int_M \left|\nabla^\Sigma \psi\right|^2f dx-\int_M
\left|D\psi\right|^2 f dx \\ =&\sum^n_{k<j} \int_{M} \left( f \left\langle \nabla^\Sigma_{e_j}\psi, e_je_k \cdot_{\text{Cl}}\nabla^\Sigma_{e_k}\psi\right\rangle dx + f \left\langle \nabla^\Sigma_{e_k} \psi, e_ke_j
\cdot_{\text{Cl}}\nabla^\Sigma_{e_j}\psi\right\rangle\right) dx\\ =&\sum^n_{\underset{j\neq k }{j,k=1}} \int_M \left(\nabla^M_{e_k}f\right)\left\langle e_k\cdot_{\text{Cl}}\psi,e_j \cdot_{\text{Cl}}\nabla^\Sigma_{e_j} \psi \right\rangle dx -\int_M
f\left\langle \psi,\mathcal{K} \psi \right\rangle dx. \end{aligned} \label{eq:step3help5}
\end{align}\tag{16}\] Inserting 10 into 16 yields \[\begin{align} \begin{aligned}
&\sum^n_{\underset{j\neq k }{j,k=1}} \int_M \left(\nabla^M_{e_k}f\right)\left\langle e_k\cdot_{\text{Cl}}\psi,e_j \cdot_{\text{Cl}}\nabla^\Sigma_{e_j} \psi \right\rangle dx -\int_M f\left\langle \psi,\mathcal{K} \psi \right\rangle dx \\
=&\sum^n_{\underset{k\neq j }{k,j=1}}\int_{M} \Big( |\psi|_{\varepsilon}^{-\frac{2}{n-1}} \left(\nabla^M_{e_j} \chi\right) \left\langle e_j \cdot_{\text{Cl}}\psi, e_k \cdot_{\text{Cl}}\nabla^\Sigma_{e_k} \psi \right\rangle\\ &- \frac{2}{n-1}
|\psi|_{\varepsilon}^{-\frac{2}{n-1}-1} \left(\nabla^M_{e_j} |\psi|_{\varepsilon}\right) \left\langle e_j \cdot_{\text{Cl}}\psi, e_k \cdot_{\text{Cl}}\nabla^\Sigma_{e_k} \psi\right\rangle \Big) dx -\int_M f \left\langle \psi, \mathcal{K}\psi \right\rangle
dx \label{eq:step3help6} \end{aligned}
\end{align}\tag{17}\] We take real parts on both sides and observe that the first and last integral on the right hand side in 17 are exactly the two last integrals in 8 . For the
integral involving \(\nabla^M_{e_j} |\psi|_{\varepsilon}\), we insert 11 into the integral and obtain \[\begin{align} \begin{aligned} -\frac{2}{n-1}&\int_M \sum_{\underset{j\neq k }{k,j=1}}^n |\psi|_{\varepsilon}^{-\frac{2}{n-1}-1}\left(\nabla |\psi|_{\varepsilon}\right) \left\langle e_j \cdot_{\text{Cl}}\psi, e_k
\cdot_{\text{Cl}}\nabla^{\Sigma}_{e_k} \psi \right\rangle \chi dx\\ =-\frac{2}{n-1}&\int_M |\psi|_{\varepsilon}^{-\frac{2}{n-1}-1} \text{Re} \left(\left\langle (\nabla^M |\psi|_{\varepsilon}),D\psi \right\rangle \right) \chi dx \\
+\frac{2}{n-1}&\int_M |\psi|_{\varepsilon}^{-\frac{2}{n-1}-1} g\left((\nabla^M|\psi|_{\varepsilon}), \text{Re}\left(\left\langle \psi, \nabla^{\Sigma}\psi \right\rangle\right)\right)\chi dx. \end{aligned} \label{eq:step3helpnew}
\end{align}\tag{18}\] We observe that the first integral in 18 is exactly the third term on the right hand side of 8 . Imposing 5 in the last integral in
18 yields \[\begin{align} |\psi|_{\varepsilon}^{-\frac{2}{n-1}-1}g\left(\nabla^M|\psi|_{\varepsilon},\text{Re}\left(\left\langle \psi, \nabla^{\Sigma} \psi \right\rangle\right)
\right)=|\psi|_{\varepsilon}^{-\frac{2}{n-1}} \left|\nabla^M |\psi|_{\varepsilon}\right|^2.
\end{align}\] To finish the proof of step 3 we note that rewriting the right hand side yields \[\begin{align} \frac{2}{n-1}|\psi|_{\varepsilon}^{-\frac{2}{n-1}} \left| \nabla^M|\psi|_{\varepsilon}\right|^2
=\frac{2(n-1)}{(n-2)^2}\left| \nabla^{M}\left( |\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\right)\right|^2
\end{align}\] and 8 follows.
For step (4) we claim \[\begin{align} \begin{aligned} \int_{M} |\psi|_{\varepsilon}^{-\frac{2}{n-1}}\left|\nabla^\Sigma\psi\right|^2dx =& \frac{2(n-1)}{(n-2)^2} \int_{M}
\left|\nabla^M\left( |\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\right) \right|^2dx \\ +&\int_{M}|\psi|_{\varepsilon}^{-\frac{2}{n-1}} \left|D\psi\right|^2 dx \\ -& \frac{2}{n-1}\int_{M} |\psi|_{\varepsilon}^{-\frac{2}{n-1}-1} \text{Re}\left(
\left\langle \left(\nabla^M |\psi|_{\varepsilon}\right)\cdot_{\text{Cl}}\psi, D\psi \right\rangle \right)dx \\ -& \int_{M}|\psi|_{\varepsilon}^{-\frac{2}{n-1}} \text{Re}\left( \left\langle \psi, \mathcal{K}\psi \right\rangle \right). \end{aligned}
\label{eq:step4}
\end{align}\tag{19}\] To see 19 we cover \(M\) by sets \(U_l\) with the properties in step (3) and choose a partition of unity \(\chi_l\) subordinate to \(U_l\). We apply 8 for \(\chi_l\) and sum over \(l\). We note that \[\begin{align} \sum_l \sum_{\underset{j\neq k}{j,k=1}}^{n}&\int_{M} |\psi|_{\varepsilon}^{-\frac{2}{n-1}}\text{Re}\left(\left\langle e_j \cdot_{\text{Cl}}\psi, e_k \cdot_{\text{Cl}}\nabla^\Sigma_{e_k} \psi\right\rangle\right)
\left(\nabla^M_{e_j} \chi_l\right) dx\\ = \sum_{\underset{j\neq k}{j,k=1}}^{n}&\int_{M} |\psi|_{\varepsilon}^{-\frac{2}{n-1}}\text{Re}\left(\left\langle e_j \cdot_{\text{Cl}}\psi, e_k \cdot_{\text{Cl}}\nabla^\Sigma_{e_k} \psi\right\rangle\right)
\left(\nabla^M_{e_j}\sum_{l} \chi_l\right) dx=0,
\end{align}\] as \(\sum_l \chi_l \equiv 1\). Collecting terms we arrive at 19 .
To finish we use 6 and 7 and observe \[\begin{align} &\left|\nabla^{\Sigma} \left( \frac{\psi}{|\psi|_{\varepsilon}^{\frac{n}{n-1}}}\right)\right| |\psi|_{\varepsilon}^2 -
\frac{1}{n} \left|-iD \left(\frac{\psi}{|\psi|_{\varepsilon}^{n/n-1}}\right)\right|^2|\psi|_{\varepsilon}^2 \\ =&\frac{\left|\nabla^{\Sigma} \psi\right|^2}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}} + \left| \nabla^M \left(
|\psi|_{\varepsilon}^{\frac{n-2}{n-1}} \right) \right|^2\left[ \left(\frac{n}{n-2}\right)^2\frac{\left|\psi\right|^2}{|\psi|_{\varepsilon}^2} - \frac{2n(n-1)}{(n-2)^2} \right] \\ +&\frac{\left|D\psi\right|^2}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}}
-\frac{1}{n} \left(\frac{n}{n-2}\right)^2\left|\nabla^M \left(|\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\right)\right|^2\frac{\left|\psi\right|^2}{|\psi|_{\varepsilon}^2} \\ +&\frac{2}{n-1}\frac{1}{|\psi|_{\varepsilon}^{\frac{2}{n-1}+1}}\text{Re} \left(
\left\langle D\psi, \nabla^M |\psi|_{\varepsilon}\cdot_{\text{Cl}}\psi \right\rangle \right).
\end{align}\] Integrating over \(M\), using 19 to express the first and last term above in terms of \(|D\psi|^2\) and \(|\nabla^M\big(|\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\big)|^2\), here also the term involving \(\mathcal{K}\) appears, and rewriting the identity yields [eq:integraliden]. ◻
In this section, we prove Theorem [thm:spin]. In Theorem [thm:mainyam] we first prove a
slightly more general version of Theorem [thm:spin]. By the means of Remark [rem:spin], Theorem [thm:spin] follows as a corollary.
The proof follows the same strategy as the proof of [1]. Since employing the Yamabe-invariant yields some technical issues, we will include the full
proof here for the sake of better readability.
Proof of Theorem [thm:mainyam]. By compactness of \(M\) and \(\psi \in L^{p}(M,\Sigma)\) for some \(p \in [\frac{2n}{n-2},\infty)\) we obtain due to Hölder’s inequality \[\begin{align} \psi \in L^{\tilde{p}}(M,\Sigma) \quad \text{for all } \tilde{p} \in [2,p]. \end{align}\] Hence, 2 , \(A\in L^n(M,TM)\) and Hölder’s inequality imply \(D\psi \in L^2(M,\Sigma)\) and Garding’s inequality, cf. Proposition [rem:gard], yields \(\psi \in H^{1}(M,\Sigma)\). By [eq:twistor95abs], we rewrite the left hand side in our integral identity and observe \[\begin{align} |T_g \varphi|^2= \sum_{j=1}^{n} \Big|\big[\nabla_{e_j} + \frac{1}{n} e_j \cdot_{\text{Cl}}D \big]\varphi\Big|^2\geq 0 \label{eq:pmainthm1} \end{align}\tag{20}\] for \(\varphi \in \Gamma(M,\Sigma)\). Making use of 20 and \(|\psi|\leq |\psi|_{\varepsilon}\), we rearrange [eq:integraliden] and obtain \[\begin{align} \frac{n-1}{n-2} \int_M \left| \nabla^{M} \left(|\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\right) \right|^2 dx\leq \frac{n-1}{n} \int_M \frac{|D\psi|^2}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}}dx - \int_M \frac{\left\langle\mathcal{K} \psi,\psi \right\rangle}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}}dx \end{align}\] After inserting 2 and \(\mathcal{K}=F^{\Sigma} +\frac{\text{scal}_g}{4}\) the inequality above can be rewritten as \[\begin{align} \begin{aligned} \frac{n-1}{n-2} \int_M \left| \nabla^{M} \left(|\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\right) \right|^2 dx\leq& \frac{n-1}{n} \int_M \frac{\left\langle \left(|A|^2\text{id} -\frac{n}{n-1} F^{\Sigma}\right)\psi,\psi\right\rangle}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}}dx \\ &-\int_M \frac{\text{scal}_g}{4} \frac{|\psi|^2}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}}dx \end{aligned} \label{eq:pmainthm2} \end{align}\tag{21}\] We define the function \(\widetilde{|\psi|}_{\varepsilon} : = |\psi|_{\varepsilon}^{\frac{n-2}{n-1}}- \varepsilon^{\frac{n-2}{n-1}}\) and observe \(\nabla^M \widetilde{|\psi|}_{\varepsilon} = \nabla^M |\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\). After adding \(\int_M \frac{\text{scal}_g}{4}\widetilde{|\psi|}^2_{\varepsilon}dx\) on both sides in 21 , we estimate the left hand side in 21 by inserting the definition of the Yamabe invariant. This yields \[\begin{align} \frac{Y(M,[g])}{4}\big(\int_{M}\widetilde{|\psi|}^{\frac{2n}{n-2}}_{\varepsilon}\big)^{\frac{n-2}{n}} \leq& \frac{n-1}{n}\int_M \frac{ \langle\left(|A|^2\text{id}- \frac{n}{n-1}F^\Sigma\right)\psi,\psi\rangle}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}} \\ &+\int_M \frac{\text{scal}_g}{4} \bigg[\widetilde{|\psi|}^2_{\varepsilon}-\frac{|\psi|^2}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}} \bigg] \end{align}\] It remains to understand the behavior of \(\varepsilon\rightarrow 0\) on both sides. We have three estimates
as \(|\psi|_{\varepsilon}\) is non-increasing in \(\varepsilon\), by monotone convergence we have \[\begin{align} \lim_{\varepsilon\rightarrow 0 } \int_M \widetilde{|\psi|}_{\varepsilon}^{\frac{2n}{n-2}} dx= \int_M |\psi|^{\frac{2n}{n-1}} dx, \end{align}\]
Hölder’s inequality and \(|\psi|_{\varepsilon}^{-\frac{2}{n-1}}|\psi|^2\leq |\psi|^{\frac{2(n-2)}{n-1}}\) imply \[\begin{align} \int_M \frac{\langle\left(|A|^2\text{id} - \frac{n}{n-1}F^\Sigma\right)\psi,\psi\rangle}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}} \leq \big\| |A|^2\text{id} - \frac{n}{n-1}F^\Sigma \big\|_{L^{\frac{n}{2}}} \big(\int_M |\psi|^{\frac{2n}{n-1}}dx\big)^{\frac{n-2}{n}}, \end{align}\]
\(\psi \in L^{\frac{2(n-2)}{n-1}}(M,\Sigma)\), dominated convergence and \(n\geq 3\) yield \[\begin{align} \lim_{\varepsilon\rightarrow 0} \int_M \frac{|\psi|_{\varepsilon}^2-|\psi|^2}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}} dx= \int_M \lim_{\varepsilon\rightarrow 0}\frac{\varepsilon^2}{(|\psi|^2+ \varepsilon^2)^{\frac{1}{n-1}}}dx=0. \end{align}\]
Thus, the assertion that \(\psi\) is nontrivial implies [eq:inequYam]. ◻
Imposing slightly stronger assumptions as in Theorem [thm:mainyam], we can prove a similar theorem for manifolds of bounded geometry. As before Theorem [thm:mainboundedgeomspin] follows as a corollary by the means of Remark [rem:spin].
Proof. The assertions \(\|A\|_{L^{\infty}}<\infty\) and \(\psi \in L^2(M,\Sigma)\) imply \(\|D\psi\|_{L^2}<\infty\) and hence, by [17] and Proposition [rem:gard], \(\psi \in H^{1,2}(M,\Sigma)\). By [21] \(H^{1,2}(M,\Sigma)\subset L^{2n/(n-2)}(M,\Sigma)\) holds for manifolds and vector bundles of bounded geometry. Thus, by interpolation of Sobolev spaces, we have \(\psi \in L^{p}(M,\Sigma)\) for \(p \in \{2n/(n-2),2n/(n-1),2(n-2)/(n-1),2\}\). The rest of the proof now follows exactly the proof of Theorem [thm:mainyam]. ◻
In this section, we will first recover the case of equality in [eq:inequYamspin] on \((\mathbb{S}^n,g_{\mathbb{S}^n})\) for odd
\(n\). Afterwards, we give a characterization of closed manifolds on which equality in [eq:inequYamspin] is possibly attained.
Throughout we assume that \((M,g,\sigma)\) is a closed Riemannian spin manifold with fixed spin structure \(\sigma\).
We first recall some facts about the case of equality on \(\mathbb{R}^n\) from [1].
We will now give a characterization of closed manifolds where equality is possibly attained in [eq:inequYamspin]. The proof follows the same strategy as the proof
of [1], we will therefore only present a sketch of the proof.
Proof. We first assume that \(|\psi|> 0\) and that \(|\psi|\) is smooth. At the end we will spend some words about the possible zeros of \(|\psi|\). We define \[\begin{align} P:= \int_M \left|T_g \left(\frac{\psi}{|\psi|^{\frac{n}{n-1}}}\right)\right|^2 \left|\psi\right|^2 dx.
\end{align}\] As \(|\psi|>0\) we can set \(\varepsilon=0\) in the integral identity [eq:integraliden] and obtain \[\begin{align} P=\frac{n-1}{n}\int_M \left| A \right|^2 \left|\psi
\right|^{\frac{2(n-2)}{n-1}}dx&-\frac{n-1}{n-2}\int_M\left| \nabla^{M}\left(|\psi|^{\frac{n-2}{n-1}}\right) \right|^2 dx\\ &-\int_M \frac{\text{scal}_g}{4} |\psi|^{\frac{2(n-2)}{n-1}}dx. \label{eq:thm95equality951}
\end{align}\tag{22}\] Rearranging 22 in terms of Hölder and Yamabe inequality yields \[\begin{align} P+R^{(1)}+R^{(2)}=S,
\end{align}\] where \[\begin{align} R^{(1)}&=\frac{n-1}{n} \left[\|A\|^2_{L^n} \left(\int_M |\psi|^{\frac{2n}{n-1}}dx \right)^{\frac{n-2}{n}}- \int_M |A|^2 |\psi|^{\frac{2(n-2)}{n-1}} dx\right], \\
R^{(2)}&=\frac{n-1}{n-2}\bigg[\int_M \left|\nabla^{M} \left(|\psi|^{\frac{n-2}{n-1}}dx\right) \right|^2 +\frac{1}{c_n}\int_M \text{scal}_g |\psi|^{\frac{2(n-2)}{n-1}}dx\\ &- Y(M,[g]) \left(\int_M
|\psi|^{\frac{2n}{n-1}}dx\right)^{\frac{n-2}{n}}\bigg]
\end{align}\] and \[\begin{align} S=\left( \frac{n-1}{n} \|A\|^2_{L^n}-\frac{Y(M,[g])}{4}\right)\left(\int_M |\psi|^{\frac{2n}{n-1}}dx\right)^{\frac{n-2}{n}}
\end{align}\] for \(c_n=\frac{4(n-1)}{n-2}\). According to the definition of the Yamabe invariant, Hölder’s inequality and Remark [rem:twistor], we have \(R^{(1)},R^{(2)},P\geq0\) and hence \(S\geq0\). Thus, \[\begin{align} S= \left(\frac{n-1}{n}
\|A\|^2_{L^n}-\frac{Y(M,[g])}{4}\right)\left(\int_M |\psi|^{\frac{2n}{n-1}}dx\right)^{\frac{n-2}{n}}\geq0
\end{align}\] and as \(\psi \neq0\) we conclude [eq:inequYamspin] again. As we assume \(\|A\|^2_{L^n}=\frac{n}{4(n-1)}Y(M,[g])\), we obtain \(S=0\) and hence \(P=R^{(1)}=R^{(2)}=0\). For \(R^{(1)}=0\), we need
equality in Hölder’s inequality and hence, \[\begin{align} |A|=\lambda |\psi|^{\frac{2}{n-1}}
\end{align}\] for some constant \(\lambda\). If \(R^{(2)}=0\), then due to the definition of the Yamabe invariant \(|\psi|^{\frac{n-2}{n-1}}\) is an
optimizer for the Yamabe problem and hence \((M,\tilde{g})\), where \(\tilde{g}= |\psi|^{\frac{4}{n-1}}g\), has constant scalar curvature \(\text{scal}_{\tilde{g}}=Y(M,[g])\). Finally, as \(P=0\), we have \(T_g\left(\frac{\psi}{|\psi|^{\frac{n}{n-1}}}\right)=0\).
Let us show that \(|\psi|\) indeed does not have any zeros. We again use the regularization \(|\psi|_{\varepsilon}\) introduced in Section 3 and rewrite
the integral identity in a similar manner as above. Now \(P=P_{\varepsilon},R^{(1)}=R^{(1)}_{\varepsilon},R^{(2)}=R^{(2)}_{\varepsilon}\) and \(S=S_{\varepsilon}\) are all dependent on \(\varepsilon\) and an additional term \(R_{\varepsilon}\) appears, where \[\begin{align} R_{\varepsilon}= \frac{n(n-1)}{(n-2)^2} \int_M \left|\nabla^M\left(
|\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\right)\right|^2 \frac{\varepsilon^2}{|\psi|_{\varepsilon}^2}dx.
\end{align}\] We directly observe that \(R_{\varepsilon}\geq0\). Moreover, we write \(S_{\varepsilon}=S^{(1)}_{\varepsilon}+S^{(2)}_{\varepsilon}\), given by \[\begin{align} S^{(1)}_{\varepsilon}=&\frac{n-1}{n} \|A\|^2_{L^n}\left(\int_M |\psi|^{\frac{2n}{n-2}}|\psi|_{\varepsilon}^{-\frac{2n}{(n-1)(n-2)}}dx \right)^{\frac{n-2}{n}}\\ &-\frac{n-1}{n-2}\frac{Y(M,[g])}{c_n} \left(\int_M
\widetilde{|\psi|}^{\frac{2n}{n-2}}_{\varepsilon}dx \right)^{\frac{n-2}{n}},\\ S^{(2)}_{\varepsilon}=& \int_M \frac{\text{scal}_g}{4} \left( \frac{|\psi|^2}{|\psi|_{\varepsilon}^{\frac{2}{n-1}}} -\widetilde{|\psi|}^2_{\varepsilon}dx \right).
\end{align}\] Hence, as \(R_\varepsilon,P_{\varepsilon},R^{(1)}_{\varepsilon},R^{(2)}_{\varepsilon}\geq0\), we have \[\begin{align} S^{(1)}_{\varepsilon}\geq -S^{(2)}_{\varepsilon}
\end{align}\] and by monotone convergence and \(\psi \in L^{\frac{2n}{n-1}}(M,\Sigma)\) we conclude \[\begin{align} \left( \frac{n-1}{n-2}\left(\|A\|^2_{L^n}- \frac{Y(M,[g])}{c_n}\right)
\left(\int_M|\psi|^{\frac{2n}{n-1}}dx \right)^{\frac{n-2}{n}} \right) &=\lim_{\varepsilon\rightarrow 0}S^{(1)}_{\varepsilon} \\ &\geq \lim_{\varepsilon\rightarrow 0}-S^{(2)}_{\varepsilon}=0.
\end{align}\] As we assume that \(\psi\) is nontrivial, we obtain \[\begin{align} \|A\|^2_{L^n}\geq\frac{n}{4(n-1)}Y(M,[g]).
\end{align}\] Imposing [eq:equality] yields \[\begin{align} \lim_{\varepsilon\rightarrow
0}R_{\varepsilon}=\lim_{\varepsilon\rightarrow0}P_{\varepsilon}=\lim_{\varepsilon\rightarrow0}R^{(1)}_{\varepsilon} =\lim_{\varepsilon\rightarrow0}R^{(2)}_{\varepsilon}=0.
\end{align}\] For our purpose particularly interesting is the term \(R^{(2)}_{\varepsilon}\) given by \[\begin{align} R^{(2)}_{\varepsilon}&= \frac{n-1}{n-2} \Big[ \int_M\left(
\left|\nabla^{M}|\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\right|^2 +\frac{1}{c_n} \text{scal}_g\widetilde{|\psi|}_\varepsilon \right)dx\\ &-\frac{Y(M,[g])}{c_n}\left(\int_M \widetilde{|\psi|}_{\varepsilon}^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\Big],
\end{align}\] where \(\widetilde{|\psi|}_{\varepsilon}=|\psi|^{\frac{n-2}{n-1}}-\varepsilon^{\frac{n-2}{n-1}}\). By monotone convergence \[\begin{align}
\left(\int_{M}\widetilde{|\psi|}^{\frac{2n}{n-2}}_{\varepsilon} \right)^{\frac{n-2}{n}}\underset{\varepsilon\rightarrow0}{\rightarrow} \left(\int_M |\psi|^{\frac{2n}{n-1}}\right)^{\frac{n-2}{n}}
\end{align}\] and \[\begin{align} \frac{1}{c_n}\int_M\text{scal}_g \widetilde{|\psi|}^2_{\varepsilon} \underset{\varepsilon\rightarrow0}{\rightarrow} \frac{1}{c_n}\int_M \text{scal}_g |\psi|^{\frac{2(n-2)}{n-1}}.
\end{align}\] The fact that \(R^{(2)}_{\varepsilon}\rightarrow 0\) implies in particular that \(R^{(2)}_{\varepsilon}\) remains bounded. Thus, \(\int_M
\left|\nabla^{M} |\psi|_{\varepsilon}^{\frac{n-2}{n-1}}\right|^2dx\) remains bounded. Thus, as for \(\varepsilon\rightarrow0\), we have \(|\psi|_{\varepsilon}^{\frac{n-2}{n-1}} \rightarrow
|\psi|^{\frac{n-2}{n-1}}\) in \(L^{1}_{\text{loc}}\), we observe that \(|\psi|^{\frac{n-2}{n-1}}\) is weakly differentiable. By [1], we obtain \[\begin{align} \int_M \left| \nabla^{M} |\psi|^{\frac{n-2}{n-1}}\right|^2dx \leq \liminf_{\varepsilon\rightarrow 0} \int_M |\nabla^M
|\psi|_{\varepsilon}^{\frac{n-2}{n-1}}|^2dx.
\end{align}\] Thus, \[\begin{align} \underset{\varepsilon\rightarrow 0}{\liminf} R^{(2)}_{\varepsilon}\geq& \int_M |\nabla^M |\psi|^{\frac{n-2}{n-1}}|^2dx + \frac{1}{c_n} \int_M \text{scal}_g
|\psi|^{\frac{2(n-2)}{n-1}} dx\\ &- \frac{Y(M,[g])}{c_n} \left(\int_M |\psi|^{\frac{2n}{n-1}} dx\right)^{\frac{n-2}{n}}.
\end{align}\] As \(\lim_{\varepsilon\rightarrow 0}R^{(2)}_{\varepsilon}=0\), we obtain \[\begin{align} \frac{Y(M,[g])}{c_n} \left( \int_M |\psi|^{\frac{2n}{n-1}}dx
\right)^{\frac{n-2}{n}}=\int_M |\nabla^M |\psi|^{\frac{n-2}{n-1}}|^2dx +\frac{1}{c_n}\int_M \text{scal}_g|\psi|^{\frac{2(n-2)}{n-1}}dx.
\end{align}\] Hence, \(|\psi|^{\frac{n-2}{n-1}}\) is an optimizer for the Yamabe problem. Thus, \(|\psi|^\frac{n-2}{n-1}>0\). ◻
Before we characterize the manifolds on which equality in [eq:inequYamspin] is possibly attained, we introduce Killing spinors.