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Geometric analysis on weighted manifolds under lower $0$-weighted Ricci curvature bounds

Yasuaki Fujitani, Yohei Sakurai

TL;DR

This work develops a geometric-analytic framework for weighted Riemannian manifolds under lower $0$-weighted Ricci curvature bounds, extending classical results to the weak setting $ ext{Ric}_f^0 geq 0$ and related bounds. The authors prove a Wang–Xia type bound for the first nonzero Steklov eigenvalue on compact manifolds with boundary, a Choi–Wang type bound for the first nonzero eigenvalue on closed $f$-minimal hypersurfaces, an ABP estimate, a Krylov–Safonov Harnack inequality, and a Brendle–type Sobolev inequality (with an isoperimetric corollary). The approach combines a Reilly-type formula for $ ext{Ric}_f^0$, a weighted Bochner framework, and Riccati-inequality techniques, along with a conformal deformation and universal-cover arguments to handle topology. Together, these results advance spectral geometry and functional-analytic inequalities in the setting of weighted manifolds with weak curvature bounds, with potential applications to mean curvature flow and non-smooth analogues of CD(K,N) spaces.

Abstract

We develop geometric analysis on weighted Riemannian manifolds under lower $0$-weighted Ricci curvature bounds. Under such curvature bounds, we prove a first non-zero Steklov eigenvalue estimate of Wang-Xia type on compact weighted manifolds with boundary, and a first non-zero eigenvalue estimate of Choi-Wang type on closed weighted minimal hypersurfaces. We also produce an ABP estimate and a Sobolev inequality of Brendle type.

Geometric analysis on weighted manifolds under lower $0$-weighted Ricci curvature bounds

TL;DR

This work develops a geometric-analytic framework for weighted Riemannian manifolds under lower -weighted Ricci curvature bounds, extending classical results to the weak setting and related bounds. The authors prove a Wang–Xia type bound for the first nonzero Steklov eigenvalue on compact manifolds with boundary, a Choi–Wang type bound for the first nonzero eigenvalue on closed -minimal hypersurfaces, an ABP estimate, a Krylov–Safonov Harnack inequality, and a Brendle–type Sobolev inequality (with an isoperimetric corollary). The approach combines a Reilly-type formula for , a weighted Bochner framework, and Riccati-inequality techniques, along with a conformal deformation and universal-cover arguments to handle topology. Together, these results advance spectral geometry and functional-analytic inequalities in the setting of weighted manifolds with weak curvature bounds, with potential applications to mean curvature flow and non-smooth analogues of CD(K,N) spaces.

Abstract

We develop geometric analysis on weighted Riemannian manifolds under lower -weighted Ricci curvature bounds. Under such curvature bounds, we prove a first non-zero Steklov eigenvalue estimate of Wang-Xia type on compact weighted manifolds with boundary, and a first non-zero eigenvalue estimate of Choi-Wang type on closed weighted minimal hypersurfaces. We also produce an ABP estimate and a Sobolev inequality of Brendle type.
Paper Structure (14 sections, 20 theorems, 93 equations)

This paper contains 14 sections, 20 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Background on curvature conditions
  3. Main theorems
  4. Organization
  5. Steklov eigenvalue estimate
  6. Reilly formula
  7. Proof of Theorem \ref{['thm:Wang-Xia']}
  8. Eigenvalue estimate on weighted minimal hypersurface
  9. Frankel property
  10. Proof of Theorem \ref{['thm:Choi-Wang']}
  11. ABP estimate
  12. Proof of Theorem \ref{['thm:ABP']}
  13. Harnack inequality
  14. Sobolev inequality

Key Result

Theorem 2.1

Let $(M,g,f)$ be a compact weighted Riemannian manifold with boundary. For $\sigma, k > 0$, we assume $\operatorname{Ric}_f^0 \geq 0$, $\mathrm{II}_{\partial M} \geq \sigma \,g_{\partial M}$ and $H_{f,\partial M} \geq k$. Then where $\lambda_{1,\partial M}$ is the first non-zero eigenvalue of the weighted Laplacian $\Delta_{f,\partial M}$ on $\bm$.

Theorems & Definitions (41)

  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3: W2
  • Proposition 2.4
  • proof
  • Proposition 2.5: LX
  • Remark 2.6
  • Theorem 2.7: W2
  • Theorem 2.8
  • proof
  • ...and 31 more