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Guofang Wei

Publications and source records attributed to Guofang Wei.

At least 19 recordsLinked to original sources
arXiv2026

A Sharp Diameter-Dependent Lower Bound for the First Nonzero Neumann Eigenvalue of Geodesic Triangles in Space Forms

We prove a sharp lower bound for the first nonzero Neumann eigenvalue of geodesic triangles of given diameter in two-dimensional space forms. The bound is given by the first positive radial Neumann eigenvalue of an one dimensional model; it is approached by degenerating isosceles triangles. When \(K>0\) and \(D=\pi/(2\sqrt K)\), equality is attained precisely by birectangular triangles. We also prove a hot-spots theorem for non-acute spherical triangles of diameter at most \(\pi/2\), and establish antisymmetry and eigenvalue monotonicity for isosceles spherical triangles of diameter \(\pi/2\).

math.DG↗
arXiv2026

Log-Concavity and Level-Set Horoconvexity of the First Eigenfunction on Horoconvex Domains in the Hyperbolic Plane

Let $\Omega\subset\mathbb H^2$ be a bounded smooth horoconvex domain and let $\psi_1>0$ be its first Dirichlet eigenfunction. We prove that \[ \operatorname{Hess}_{\mathbb H^2}(-\log\psi_1)>0 \] throughout $\Omega$, with no restriction on the diameter or the first eigenvalue. The proof is by contradiction. A degenerate Hessian would yield a shifted translation Killing derivative with a singular interior zero. Then the boundary-zero theorem of Grossi and Provenzano shows that the shifted Killing derivative has exactly two zeros on the boundary. A nodal-domain argument on the surface rules this out. As an application we prove that every superlevel set of $\psi_1$ is horoconvex: every level curve has geodesic curvature at least $1$. The Hessian bound makes the shifted construction available for Killing fields with nonvanishing rotation part, and yields the pointwise inequality $|(\operatorname{Hess} u)^{-1}J\nabla u|\le1$ for $u=-\log\psi_1$, where $J$ is rotation by $\pi/2$; a boundary-zero count for translation fields with arbitrary axis completes the argument.

math.DG↗
arXiv2026

A Large-Diameter Fundamental-Gap Lower Bound for Horoconvex Domains

We prove a large-diameter fundamental-gap lower bound for compact horoconvex domains in real hyperbolic space of curvature \(-1\). The geometric part reduces large horoconvex domains to a fixed-width radial-height problem in all dimensions. The analytic part proves the needed radial-height theorem by comparing the low-energy Dirichlet form with a limiting angular operator on the sphere, while the radial complement is separated by a one-dimensional branch gap and endpoint Green estimates. The result gives the polynomial \(D^{-3}\) scale matching the Nguyen--Stancu--Wei large-diameter upper bound.

math.DG↗
arXiv2025

Super Log-concavity of the First Eigenfunctions for Horo-convex Domains in Hyperbolic Space

In this paper, we prove that the first eigenfunction of the Laplacian for a horo-convex domain $\Omega\subset\mathbb H^n$ is super log-concave when $\text{diam}(\Omega)$ is not large. Our result is optimal in the sense that there are counterexamples %are constructed for the cases when $\Omega$ is not horo-convex or when $\text{diam}(\Omega)$ is large respectively

math.AP↗
arXiv2024

Singular metrics with nonnegative scalar curvature and RCD

We show that a uniformly Euclidean metric with isolated singularity on $M^n = T^n \# M_0$, where $4\leq n\leq 7$ or $n\geq 4$, $M_0$ spin, and nonnegative scalar curvature on the smooth part is Ricci flat and extends smoothly over the singularity. This confirms Schoen's Conjecture in these cases. The key to the proof is to show that the space has nonnegative synthetic Ricci curvature, i.e., an $RCD(0, n)$ space. Our result also holds when the singular set consists of a finite union of submanifolds (of possibly different dimensions) intersecting transversally under additional assumption on the co-dimension and the location of the singular set.

math.DG↗
arXiv2024

Volume entropy and rigidity for RCD-spaces

We develop the barycenter technique of Besson--Courtois--Gallot so that it can be applied on RCD metric measure spaces. Given a continuous map $f$ from a non-collapsed RCD$(-(N-1),N)$ space $X$ without boundary to a locally symmetric $N$-manifold we show a version of BCG's entropy-volume inequality. The lower bound involves homological and homotopical indices which we introduce. We prove that when equality holds and these indices coincide $X$ is a locally symmetric manifold, and $f$ is homotopic to a Riemannian covering whose degree equals the indices. Moreover, we show a measured Gromov--Hausdorff stability of $X$ and $Y$ involving the homotopical invariant. As a byproduct, we extend a Lipschitz volume rigidity result of Li--Wang to RCD$(K,N)$ spaces without boundary. Finally, we include an application of these methods to the study of Einstein metrics on $4$-orbifolds.

math.DG↗
arXiv2024

Finite Diffeomorphism Theorem for manifolds with lower Ricci curvature and bounded energy

In this paper we prove that the space $\cM(n,\rv,D,Λ):=\{(M^n,g) \text{ closed }: ~~\Ric\ge -(n-1),~\Vol(M)\ge \rv>0, \diam(M)\le D \text{ and } \int_{M}|\Rm|^{n/2}\le Λ\}$ has at most $C(n,\rv,D,Λ)$ many diffeomorphism types. This removes the upper Ricci curvature bound of Anderson-Cheeger's finite diffeomorphism theorem in \cite{AnCh}. Furthermore, if $M$ is Kähler surface, the Riemann curvature $L^2$ bound could be replaced by the scalar curvature $L^2$ bound.

math.DG↗
arXiv2023

Probabilistic Method to Fundamental gap problems on the sphere

We provide a probabilistic proof of the fundamental gap estimate for Schr\"odinger operators in convex domains on the sphere, which extends the probabilistic proof of F. Gong, H. Li, and D. Luo for the Euclidean case. Our results further generalize the results achieved for the Laplacian by S. Seto, L. Wang, and G. Wei, as well as by C. He, G. Wei, and Qi S. Zhang. The essential ingredient in our analysis is the reflection coupling method on Riemannian manifolds.

math.PR↗
arXiv2023

Modulus of Concavity and Fundamental Gap Estimates on Surfaces

The fundamental gap of a domain is the difference between the first two eigenvalues of the Laplace operator. In a series of recent and celebrated works, it was shown that for convex domains in $\mathbb R^n$ and $\mathbb S^n$ with Dirichlet boundary condition the fundamental gap is at least $\frac{3 π^2}{D^2}$ where $D$ denotes the diameter of the domain. The key to these results is to establish a strong concavity estimate for the logarithm of the first eigenfunction. In this article, we prove corresponding log-concavity and fundamental gap estimates for surfaces with non-constant positive curvature via a two-point maximum principle. However, the curvature not being constant greatly increases the difficulty of the computation.

math.DG↗
arXiv2022

Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature

We study the log-concavity of the first Dirichlet eigenfunction of the Laplacian for convex domains. For positively curved surfaces satisfying a condition involving the curvature and its second derivatives, we show that the first eigenfunction is strongly log-concave. Previously, for general convex domains, the log-concavity of the first eigenfunctions were only known when lying in $\mathbb{R}^n$ and $\mathbb{S}^n$. Using this estimate, we establish lower bounds on the fundamental gap of such regions. Furthermore, we study the behavior of these estimates under Ricci flow and other deformations of the metric.

math.DG↗
arXiv2022

Singular Weyl's law with Ricci curvature bounded below

We establish two surprising types of Weyl's laws for some compact $\mathrm{RCD}(K, N)$/Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for $\mathrm{RCD}(K,N)$ spaces. Our results depends crucially on analyzing and developing important properties of the examples constructed by the last two authors, showing them isometric to the $\alpha$-Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures by Cheeger-Colding and by Kapovitch-Kell-Ketterer.

math.DG↗
arXiv2022

Ricci Flow and Gromov Almost Flat Manifolds

We employ the Ricci flow to derive a new theorem about Gromov almost flat manifolds, which generalizes and strengthens the celebrated Gromov--Ruh Theorem. In our theorem, the condition $diam^2 |K| \leq ε_n$ in the Gromov--Ruh Theorem is replaced by the substantially weaker condition $\|Rm\|_{n/2}$ $ C_S^2 \leq \varepsilon_n$.

math.DG↗
arXiv2021

Improved relative volume comparison for integral Ricci curvature and applications to volume entropy

We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in \cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci curvature which substantially improves an earlier estimate in \cite{Au2} and give an application on the algebraic entropy of its fundamental group. We also extend the almost minimal volume rigidity of \cite{BBCG} to integral Ricci curvature.

math.DG↗
arXiv2021

Integral Ricci curvature and the mass gap of Dirichlet Laplacians on domains

We obtain a fundamental gap estimate for classes of bounded domains with quantitative control on the boundary in a complete manifold with integral bounds on the negative part of the Ricci curvature. This extends the result of \cite{Oden-Sung-Wang99} to $L^p$-Ricci curvature assumptions, $p>n/2$. To achieve our result, it is shown that the domains under consideration are John domains, what enables us to obtain an estimate on the first nonzero Neumann eigenvalue, which is of independent interest.

math.DG↗
arXiv2021

Examples of Ricci limit spaces with non-integer Hausdorff dimension

We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding about collapsing Ricci limit spaces.

math.DG↗
arXiv2021

The fundamental gap of horoconvex domains in $\mathbb H^n$

We show that, for horoconvex domains in the hyperbolic space, the product of their fundamental gap with the square of their diameter has no positive lower bound. The result follows from the study of the fundamental gap of geodesic balls as the radius goes to infinity. In the process, we improve the lower bound for the first eigenvalue of balls in hyperbolic space.

math.DG↗