[Paper Review] Rigidity of manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound

Yohei Sakurai|arXiv (Cornell University)|Jun 10, 2015

Geometric Analysis and Curvature Flows34 references3 citations

TL;DR

This paper establishes rigidity theorems for Riemannian manifolds with boundary under a lower Bakry-Émery Ricci curvature bound and a lower $f$-mean curvature bound. Using weighted comparison geometry and spectral analysis, it proves that the inscribed radius is bounded above by the radius of a model space, with equality implying isometry to that model. It further derives sharp lower bounds for the first eigenvalue of the weighted $p$-Laplacian, generalizing classical results to the weighted setting with boundary.

ABSTRACT

We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundaries, and various splitting theorems. We also obtain rigidity results for the smallest Dirichlet eigenvalues for the weighted p-Laplacians.

Motivation & Objective

• To extend classical rigidity results for manifolds with boundary to the weighted setting with Bakry-Émery Ricci curvature bounds.
• To establish sharp upper bounds on the inscribed radius of a manifold with boundary under curvature and $f$-mean curvature conditions.
• To derive volume growth and spectral rigidity results for the weighted $p$-Laplacian on such manifolds.
• To generalize Heintze-Karcher and Kasue-type rigidity theorems to the weighted case with smooth weight function $f$.
• To characterize equality cases in the inscribed radius and eigenvalue bounds, showing isometry to model spaces.

Proposed method

• Define the Bakry-Émery Ricci curvature $\operatorname{Ric}^N_f \geq (N-1)\kappa$ and $f$-mean curvature $H_{f,\partial M} \geq (N-1)\lambda$ on a complete Riemannian manifold with boundary.
• Introduce the model space $B^n_{\kappa,\lambda}$, a geodesic ball in the space form $M^n_\kappa$ with constant $f$-mean curvature $(n-1)\lambda$, and define its radius $C_{\kappa,\lambda}$.
• Use the Jacobi equation $\phi''(t) + \kappa\phi(t) = 0$ with initial conditions $\phi(0) = 1$, $\phi'(0) = -\lambda$ to define $s_{\kappa,\lambda}(t)$, whose first positive zero gives $C_{\kappa,\lambda}$.
• Establish comparison theorems via the weighted measure $m_f = e^{-f} \operatorname{vol}_g$, analyzing the weighted distance function $\rho_{\partial M}(p) = d_M(p, \partial M)$.
• Derive isoperimetric-type inequalities: $m_f(\Omega) \leq m_{f,\partial\Omega}(\partial\Omega) \cdot C(N,\kappa,\lambda,D)$ for domains $\Omega$ with $\partial\Omega \cap \partial M = \emptyset$.
• Apply the weighted Poincaré inequality and Hölder’s inequality to the Rayleigh quotient to obtain lower bounds on the first eigenvalue $\mu_{f,1,p}(M)$ of the weighted $p$-Laplacian.

Experimental results

Research questions

• RQ1 What is the sharp upper bound on the inscribed radius $D(M, \partial M)$ under a lower Bakry-Émery Ricci curvature and $f$-mean curvature bound?
• RQ2 Under what conditions does equality in the inscribed radius bound imply that the manifold is isometric to a model geodesic ball?
• RQ3 What are the sharp lower bounds for the first eigenvalue of the weighted $p$-Laplacian under curvature and boundary conditions?
• RQ4 How does the volume growth of metric neighborhoods of the boundary behave under these curvature bounds?
• RQ5 What spectral rigidity results hold in the $N=\infty$ case with non-negative Bakry-Émery Ricci curvature and $f$-mean curvature?

Key findings

• The inscribed radius satisfies $D(M, \partial M) \leq C_{\kappa,\lambda}$, where $C_{\kappa,\lambda}$ is the radius of a geodesic ball in the space form $M^n_\kappa$ with constant $f$-mean curvature $(n-1)\lambda$.
• Equality in the inscribed radius bound holds if and only if $(M,d_M)$ is isometric to $B^n_{\kappa,\lambda}$ and $N=n$, with $f$ constant on $M$.
• For $N \in [n,\infty)$, the first eigenvalue of the weighted $p$-Laplacian satisfies $\mu_{f,1,p}(M) \geq (p \cdot C(N,\kappa,\lambda,D))^{-p}$, where $D = D(M, \partial M)$.
• In the $N=\infty$ case, $\mu_{f,1,p}(M) \geq (pD)^{-p}$, with equality if and only if $D=\infty$, $\partial M$ compact, and $M$ isometric to $[0,\infty) \times_{\kappa,\lambda} \partial M$.
• When $\kappa < 0$ and $\lambda = \sqrt{|\kappa|}$, the constant $C(N,\kappa,\lambda,D)$ simplifies to $\left((N-1)\lambda\right)^{-1} \left(1 - e^{-(N-1)\lambda D}\right)$, which increases to $((N-1)\lambda)^{-1}$ as $D \to \infty$.
• Equality in the eigenvalue bound implies that $f \circ \gamma_x(t) = f(x) + (N-n)\lambda t$ along radial geodesics, confirming the model space structure.