May 27, 2026
In this short note, we establish an upper bound for the conjugate radius of an open \(n\)-dimensional Riemannian manifold under a scalar curvature lower bound and a bottom-of-spectrum upper bound. As a consequence, if \(\lambda_{0}(M)=0\) and scalar curvature \(\ge n(n-1)\), then the conjugate radius \(\le \pi\).
The conjugate radius measures the extend to which geodesics remain free of conjugate points. It is naturally tied to the Jacobi equation: along a geodesic \(\gamma\), Jacobi fields satisfy a second-order differential equation whose coefficients are determined by the sectional curvatures containing \(\dot{\gamma}\). Thus, the conjugate radius appeaars at first to be governed by sectional curvature. A remarkable theorem of Leon Green shows, however, that it is also constrained by scalar curvature in the average sense. In 1963, Green [1] proved the following result:
Theorem 1 (Green’s Theorem). Let \(M\) be an \(n\)-dimensional closed Riemannian manifold. If the conjugate radius of \(M\) is bounded from below by \({\rm conj}(M)\geq a\), then \[\frac{1}{{\rm Vol}(M)}\int_M{\rm scal}(p)~{\rm dvol}_M(p)\leq n(n-1)\frac{\pi^2}{a^2},\] with equality if and only if \(M\) has constant positive sectional curvature \(\pi^2/a^2\).
For complete noncompact Riemannian manifolds, it remains open whether a scalar curvature lower bound alone forces an upper bound on the conjugate radius.
In 2022, Zhu [2] established, among other results, the open manifold case under the additional assumption of non-negative Ricci curvature:
Theorem 2 (Zhu). Let \(M\) be an \(n\)-dimensional open Riemannian manifold satisfying \[\mathrm{scal}_M\geq n(n-1) \quad \text{and }\;{\rm Ric}_M\geq0,\] then the conjugate radius of \(M\) satisfy: \[{\rm conj}(M)\leq\pi.\]
In 2025, Kwong [3] obtained a related result under a different curvature together with the assumption of finite volume:
Theorem 3 (Kwong). Let \(M\) be an \(n\)-dimensional complete Riemannian with finite volume. Assume that
\(\int_{SM}{\rm Ric}_M^-(p;v)~d\mu(p,v)\) is finite, where \({\rm Ric}_M^-(p;v)=\max\{-{\rm Ric}_M(p;v),0\}\) denotes the negative part of the Ricci curvature and \(SM\) denotes the unit sphere bundle over \(M\);
\(\int_M{\rm scal}(p)~{\rm dvol}_M(p)\geq n(n-1){\rm Vol}(M)\).
Then the conjugate radius of \(M\) satisfies \[{\rm conj}(M)\leq\pi,\] with equality if and only if \(M\) has constant sectional curvature equal to \(1\).
In this paper we prove:
Theorem 4 (Main theorem). Let \(M\) be an \(n\)-dimensional open Riemannian manifold with \({\rm scal}_M\geq n(n-1)\) and \(\lambda_{0}(M)<n\). Then the conjugate radius of \(M\) satisfies \[{\rm conj}(M)\leq\frac{\pi{\sqrt{1-\lambda_0(M)/n}}}{,}\] where \(\lambda_{0}(M)\) denotes the bottom of the spectrum of the Laplacian on \(M\). In particular, if in addition \(\lambda_0(M)=0\), then \[{\rm conj}(M)\leq\pi.\]
Our estimate gives a finite conjugate-radius bound precisely in the range \(\lambda_{0}<n\). This places the result in the broader contex of comparison results relating curvature assumptions to upper bounds for the bottom of the spectrum.
Also, we note that either \({\rm Ric}_M\geq0\) or \({\rm Vol}(M)<\infty\) implies that the bottom of the spectrum of \(M\) vanishes, see proposition 1 and 2. Thus, our theorem may be viewed as a spectral version of Zhu’s or Kwong’s conjugate radius estimates.
Let \(M\) be a complete Riemannian manifold. Recall that the bottom of the spectrum of \(M\) is defined by \[\lambda_0(M):=\inf\left\{\left.\frac{\int_M|\nabla f|^2~{\rm dvol}_M}{\int_M f^2~{\rm dvol}_M}\right|f\in C_c^\infty(M),f\not\equiv0\right\}.\] The quantity inside the infimum is called the Rayleigh quotient. If \(M\) is compact, then every smooth function on \(M\) has compact support, so testing the Rayleigh quotient on constant functions yields \(\lambda_0(M)=0\). In this case, the more meaningful quantity is the first nonzero eigenvalue \(\lambda_1(M)\). By contrast, if \(M\) is noncompact, \(\lambda_0(M)\) may be vanish (for example, in Euclidean space) or be positive (for example, in hyperbolic space).
Now we show that either \({\rm Ric}_M\geq0\) or \({\rm Vol}(M)<\infty\) implies \(\lambda_0(M)=0\).
Proposition 1. If \(M\) is an open Riemannian manifold with \({\rm Ric}_M\geq0\), then \(\lambda_0(M)=0\).
Proof. Fix a point \(p\in M\). For each \(R>0\), define \[\varphi_R(x)=\psi\!\left(\frac{d(x,p)}{R}\right),\] where \(\psi\in C^\infty(\mathbb{R})\) satisfies \[\psi(t)=1 \quad \text{for } t\leq1,\] \[\psi(t)=0 \quad \text{for } t\geq2,\] \[0<\psi(t)<1 \quad \text{for } 1<t<2,\] and \(|\psi'|\leq C\). Then \(\varphi_R\in{\rm Lip}_c(M)\) with \(\varphi_R\equiv 1\) on \(B(p,R)\), \(\varphi_R\equiv 0\) on \(M\setminus B(p,2R)\), and \(|\nabla\varphi_R|\leq C/R\) almost everywhere. Consequently, \[\int_{\{\nabla\varphi_R~{\rm exists}\}}|\nabla\varphi_R|^2~{\rm dvol}_M\leq\frac{C^2}{R^2}{\rm Vol}(B(p,2R)\backslash B(p,R))\] while \[\int_M\varphi_R^2~{\rm dvol}_M\geq{\rm Vol}(B(p,R)).\] Hence the Rayleigh quotient satisfies \[\frac{\int|\nabla\varphi_R|^2}{\int\varphi_R^2}\leq\frac{C^2}{R^2}\cdot \frac{{\rm Vol}(B(p,2R))-{\rm Vol}(B(p,R))}{{\rm Vol}(B(p,R))} \leq\frac{C^2}{R^2}\cdot\frac{{\rm Vol}(B(p,2R))}{{\rm Vol}(B(p,R))}.\] Since \({\rm Ric}_M\geq0\), the Bishop–Gromov volume comparison theorem implies that the function \(r\mapsto{\rm Vol}(B(p,r))/\omega_nr^n\) is non-increasing. Therefore, \[\frac{{\rm Vol}(B(p,2R))}{(2R)^n}\leq\frac{{\rm Vol}(B(p,R))}{R^n}, \quad {\rm i.e.,} \frac{{\rm Vol}(B(p,2R))}{{\rm Vol}(B(p,R))}\leq 2^n.\] Substituting this estimate gives \[\frac{\int|\nabla\varphi_R|^2}{\int\varphi_R^2}\leq\frac{C^2 2^n}{R^2}\to 0 ~ \text{as } ~ R\to\infty.\] Since compactly supported Lipschitz functions can be approximated in \(W^{1,2}\) by functions in \(C^{\infty}_{c}(M)\), the above Rayleigh quotient estimates are admissible in the definition of \(\lambda_{0}(M)\). Therefore \(\lambda_0(M)=0\). ◻
Proposition 2. If \(M\) is an open Riemannian manifold with finite volume, then \(\lambda_0(M)=0\).
Proof. As in the proof of Proposition 1, \[\frac{\int|\nabla\varphi_R|^2}{\int\varphi_R^2} \leq\frac{C^2}{R^2}\cdot\frac{{\rm Vol}(B(p,2R))-{\rm Vol}(B(p,R))}{{\rm Vol}(B(p,R))}.\] Since \({\rm Vol}(B(p,R))\to{\rm Vol}(M)<\infty\) as \(R\to\infty\), we have \[{\rm Vol}(B(p,2R))-{\rm Vol}(B(p,R))\to{\rm Vol}(M)-{\rm Vol}(M)=0,\] therefore \(\lambda_0(M)=0\). ◻
Let \[l:=\frac{\pi{\sqrt{1-\frac{\lambda_0(M)}{n}}}}{.}\] Suppose, to the contrary, that \({\rm conj}(M)>l\). Choose \(a\in(l,{\rm conj}(M))\), and let \(f\in C_c^\infty(M)\) be nonzero. For each \((p,v)\in SM\), let \(\gamma(t)=\exp_p(tv)\) for \(0\leq t\leq a\). Choose parallel vector fields \(E_1(t), \cdots, E_{n-1}(t)\) along \(\gamma\) such that \[\{\gamma',E_1,\ldots,E_{n-1}\}\] forms an orthonormal frame. Define the variational vector field \[V_i(t)=f(\gamma(t))\sin\left(\frac{\pi t}{a}\right)E_i(t).\] Then \(V_i(0)=V_i(a)=0\). Since \(a< {\rm conj}(M)\), the index form is nonnegative: \[\begin{align} 0&\leq I(V_i,V_i)=\int_0^a\left(|\nabla_{\gamma'}V_i|^2-\langle R(\gamma',V_i)V_i,\gamma'\rangle\right)dt\\ &=\int_0^a\left[(f')^2\sin^2\left(\frac{\pi t}{a}\right)+2ff'\frac{\pi{a}}{\sin}\left(\frac{\pi t}{a}\right)\cos\left(\frac{\pi t}{a}\right)+f^2\frac{\pi^2}{a^2}\cos^2\left(\frac{\pi t}{a}\right)\right]dt\\ &\quad-\int_0^af^2\sin^2\left(\frac{\pi t}{a}\right)K(\gamma', E_i)dt, \end{align}\] where, for brevity, we write \(f\) for \(f\circ\gamma\) and \(f'\) for \((f\circ\gamma)'\). Here we use \[K(X, Y):=\left\langle R(X, Y)Y, X \right\rangle\] to denote the sectional curvature spanned by the orthonormal pair \(X, Y\). Hence the Ricci curvature can be written as \(\mathrm{Ric}(X, X)= \sum_{i=1}^{n-1} K(X, E_{i})\). Integration by parts yields \[\begin{align} &\quad\int_0^a2ff'\frac{\pi{a}}{\sin}\left(\frac{\pi t}{a}\right)\cos\left(\frac{\pi t}{a}\right)dt\\ &=\int_0^a(f^2)'\frac{\pi{a}}{\sin}\left(\frac{\pi t}{a}\right)\cos\left(\frac{\pi t}{a}\right)dt\\ &=\left.\left[f^2\frac{\pi{a}}{\sin}\left(\frac{\pi t}{a}\right)\cos\left(\frac{\pi t}{a}\right)\right]~\right|_{t=0}^{t=a} -\int_0^af^2\frac{\pi{a}}{\left}[\sin\left(\frac{\pi t}{a}\right)\cos\left(\frac{\pi t}{a}\right)\right]'dt\\ &=\int_0^af^2\frac{\pi^2}{a^2}\left[\sin^2\left(\frac{\pi t}{a}\right)-\cos^2\left(\frac{\pi t}{a}\right)\right]dt. \end{align}\] Substituting this gives \[\int_0^a\left[(f')^2\sin^2\left(\frac{\pi t}{a}\right)+f^2\frac{\pi^2}{a^2}\sin^2\left(\frac{\pi t}{a}\right) -f^2\sin^2\left(\frac{\pi t}{a}\right)K(\gamma',E_i)\right]dt\geq0.\] Summing over \(i=1,\cdots,n-1\) yields \[\label{eq2461} \begin{align} &\quad\int_0^af^2\sin^2\left(\frac{\pi t}{a}\right){\rm Ric}(\gamma')dt\\ &\leq(n-1)\frac{\pi^2}{a^2}\int_0^a f^2\sin^2\left(\frac{\pi t}{a}\right)dt +(n-1)\int_0^a\langle\nabla f,\gamma'(t)\rangle^2\sin^2\left(\frac{\pi t}{a}\right)dt. \end{align}\tag{1}\] Recall that, for each \(t\in\mathbb{R}\), the geodesic flow is defined by: \[\varphi_t:TM\to TM, \quad \varphi_t(q,w)=(\sigma_w(t),\sigma_w'(t)),\] where \(w\in T_qM\) and \(\sigma_w(t)=\exp_q(tw)\). In terms of the geodesic flow \(\varphi_t\) rather than \(\gamma\), 1 becomes \[\label{eq2462} \begin{align} &\quad\int_0^af(\pi_1(\varphi_t(p,v)))^2\sin^2\left(\frac{\pi t}{a}\right){\rm Ric}(\varphi_t(p,v))dt\\ &\leq(n-1)\frac{\pi^2}{a^2}\int_0^a f(\pi_1(\varphi_t(p,v)))^2\sin^2\left(\frac{\pi t}{a}\right)dt\\ &\quad+(n-1)\int_0^a\langle\nabla f,\pi_2(\varphi_t(p,v))\rangle^2\sin^2\left(\frac{\pi t}{a}\right)dt, \end{align}\tag{2}\] where \(\pi_1:(p,v)\mapsto p\) and \(\pi_2:(p,v)\mapsto v\) are projections. This holds for all \((p, v)\in SM\). Now integrate 2 over the unit tangent bundle \(SM\). The Sasaki metric on \(TM\) induces a measure \(d\mu\) on \(SM\) with \(d\mu={\rm dvol}_Md\omega_{n-1}\) called Liouville measure, where \(d\omega_{n-1}\) is the volume element of \(S^{n-1}\). By the invariance of \(d\mu\) under the geodesic flow, we have \[\int_{SM}h~d\mu=\int_{SM}h\circ\varphi_t~d\mu\] for any \(h\in C_c^\infty(SM)\) and \(t\in\mathbb{R}\). We now integrate the three terms of 2 separately. Integrating the first term over \(SM\), we obtain \[\begin{align} &\quad\int_{SM}\left(\int_0^af(\pi_1(\varphi_t(p,v)))^2\sin^2\left(\frac{\pi t}{a}\right){\rm Ric}(\varphi_t(p,v))dt\right)d\mu(p,v)\\ &=\int_0^a\left(\int_{SM}f(\pi_1(\varphi_t(p,v)))^2{\rm Ric}(\varphi_t(p,v))d\mu(p,v)\right)\sin^2\left(\frac{\pi t}{a}\right)dt\\ &=\int_0^a\sin^2\left(\frac{\pi t}{a}\right)dt\int_{SM}f(\pi_1(p,v))^2{\rm Ric}(p;v)d\mu(p,v)\\ &=\frac{a}{2}\int_M\left(\int_{S_{p}M}f(\pi_1(p,v))^2{\rm Ric}(p;v)d\omega_{n-1}(v)\right){\rm dvol}_M(p)\\ &=\frac{a}{2}\int_Mf(p)^2\left(\int_{S_{p}M}{\rm Ric}(p;v)d\omega_{n-1}(v)\right){\rm dvol}_M(p)\\ &=\frac{a\omega_{n-1}}{2n}\int_Mf(p)^2{\rm scal}_M(p)~{\rm dvol}_M(p), \end{align}\] where the order of integration can be interchanged because \(f\) has compact support. Similarly, integrating the second term in 2 gives, \[\int_{SM}\left(\int_0^a f(\pi_1(\varphi_t(p,v)))^2\sin^2\left(\frac{\pi t}{a}\right)dt\right)d\mu(p,v) =\frac{a\omega_{n-1}}{2}\int_Mf(p)^2~{\rm dvol}_M(p).\] For the third term of 2 , we obtain, \[\begin{align} &\quad\int_{SM}\left(\int_0^a\langle\nabla f,\pi_2(\varphi_t(p,v))\rangle^2\sin^2\left(\frac{\pi t}{a}\right)dt\right)d\mu(p,v)\\ &=\frac{a}{2}\int_M\left(\int_{S_{p}M}\langle\nabla f,\pi_2(p,v)\rangle^2d\omega_{n-1}(v)\right){\rm dvol}_M(p)\\ &=\frac{a\omega_{n-1}}{2n}\int_M|\nabla f(p)|^2~{\rm dvol}_M(p). \end{align}\] Substituting these gives \[\begin{align} &\quad\frac{a\omega_{n-1}}{2n}\int_Mf^2{\rm scal}_M~{\rm dvol}_M\\ &\leq(n-1)\frac{\pi^2}{2a}\omega_{n-1}\int_Mf^2~{\rm dvol}_M+(n-1)\frac{a\omega_{n-1}}{2n}\int_M|\nabla f|^2~{\rm dvol}_M. \end{align}\] Simplifying, we have \[\int_Mf^2{\rm scal}_M~{\rm dvol}_M\leq n(n-1)\frac{\pi^2}{a^2}\int_Mf^2~{\rm dvol}_M+(n-1)\int_M|\nabla f|^2~{\rm dvol}_M.\] Since \({\rm scal}_M\geq n(n-1)\), we have \[\int_Mf^2{\rm scal}_M~{\rm dvol}_M\geq n(n-1)\int_Mf^2~{\rm dvol}_M,\] which yields \[n\int_Mf^2~{\rm dvol}_M\leq n\frac{\pi^2}{a^2}\int_Mf^2~{\rm dvol}_M+\int_M|\nabla f|^2~{\rm dvol}_M,\] i.e., \[\frac{\int_M|\nabla f|^2~{\rm dvol}_M}{\int_Mf^2~{\rm dvol}_M}\geq n\left(1-\frac{\pi^2}{a^2}\right).\] Since \(a>l\), we have \[n\left(1-\frac{\pi^2}{a^2}\right)>n\left(1-\frac{\pi^2}{l^2}\right)=\lambda_0(M).\] Since \(f\in C_c^\infty(M)\setminus \{0\}\) was arbitrary, we obtain \[\lambda_0(M)\geq n\left(1-\frac{\pi^2}{a^2}\right)>\lambda_0(M),\] which is a contradiction. Hence \({\rm conj}(M)\leq l\).
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