About

Guofang Wei is a Professor in the Mathematics Department at the University of California, Santa Barbara. She earned her Ph.D. in 1989 from Stony Brook University and her B.S. in 1985 from Zhejiang University, China. Her primary research field is differential geometry and geometric analysis, which closely relates to partial differential equations, topology, Lie groups, and general relativity. Her research efforts have been concentrated on global Riemannian geometry, focusing on the interaction of curvature with the underlying geometry and topology. Specific topics of interest include the study of fundamental groups, comparison geometry, manifolds with integral curvature bounds, spaces with weak curvature bounds, and the eigenvalue of the Laplacian. In addition to her research, Professor Wei is actively involved in teaching, providing all course materials to students enrolled or waitlisted via UCSB Canvas. She has supervised several Ph.D. students and serves on editorial boards for several mathematical journals, including the Proceedings of the American Mathematical Society (2014-2022), the Journal of Topology and Analysis (since 2019), and the Annals of Global Analysis and Geometry (since 2020).

Research topics

• Mathematics
• Mathematical analysis
• Geometry
• Pure mathematics
• Artificial Intelligence
• Combinatorics
• Physics
• Computer Science
• Statistical physics
• Quantum mechanics
• Geology
• Discrete mathematics
• Algorithm

Selected publications

• Examples of open manifolds with positive Ricci curvature and non-proper Busemann functions

  American Journal of Mathematics · 2026-01-23

  articleSenior author

  abstract: We give the first example of an open manifold with positive Ricci curvature and a non-proper Busemann function at a point. This provides counterexamples to the longtime well-known open question of whether the Busemann function at a point of an open manifold with nonnegative Ricci curvature is proper.

  PublisherDOI

• Singular metrics with non-negative scalar curvature and RCD

  Communications in Contemporary Mathematics · 2026-02-20

  articleSenior author

  We show that a uniformly Euclidean metric with isolated singularity on closed [Formula: see text], where [Formula: see text] or [Formula: see text], [Formula: see text] spin, and non-negative scalar curvature on the smooth part is flat and extends smoothly over the singularity. This confirms Schoen’s Conjecture in these cases. The novel approach here, which is the key to the proof, is to show that the space has non-negative synthetic Ricci curvature, i.e. an [Formula: see text] 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.

  PublisherDOI

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

  ArXiv.org · 2025-10-15

  preprintOpen access1st authorCorresponding

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

  PublisherOA PDFDOI

• Log-concavity and fundamental gaps on surfaces of positive curvature

  Communications in Analysis and Geometry · 2025-01-01 · 2 citations

  articleSenior author
  PublisherDOI

• Spaces with Ricci Curvature Lower Bounds

  2024-12-31

  book-chapterSenior author
  PublisherDOI

• Finite diffeomorphism theorem for manifolds with lower Ricci curvature and bounded energy

  Mathematische Zeitschrift · 2024-11-30

  articleOpen accessSenior authorCorresponding

  Abstract In this paper we prove that the space $$\mathcal {M}(n,\textrm{v},D,\Lambda ):=\{(M^n,g) \text { closed }: ~~\textrm{Ric}\ge -(n-1),~\textrm{Vol}(M)\ge \textrm{v}&gt;0, \text {diam}(M)\le D \text { and } \int _{M}|\textrm{Rm}|^{n/2}\le \Lambda \}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mtext>v</mml:mtext> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace/> <mml:mtext>closed</mml:mtext> <mml:mspace/> <mml:mo>:</mml:mo> <mml:mspace/> <mml:mspace/> <mml:mtext>Ric</mml:mtext> <mml:mo>≥</mml:mo> <mml:mo>-</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mtext>Vol</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≥</mml:mo> <mml:mtext>v</mml:mtext> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mtext>diam</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>D</mml:mi> <mml:mspace/> <mml:mtext>and</mml:mtext> <mml:mspace/> <mml:msub> <mml:mo>∫</mml:mo> <mml:mi>M</mml:mi> </mml:msub> <mml:mo>|</mml:mo> <mml:mtext>Rm</mml:mtext> <mml:msup> <mml:mo>|</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>≤</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> has at most $$C(n,\textrm{v},D,\Lambda )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mtext>v</mml:mtext> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> many diffeomorphism types. This removes the upper Ricci curvature bound of Anderson-Cheeger’s finite diffeomorphism theorem in Anderson and Cheeger (Geom Funct Anal 1(3):231–252, 1991). Furthermore, if M is Kähler surface, the Riemann curvature $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> bound could be replaced by the scalar curvature $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> bound.

  PublisherOA PDFDOI

• Probabilistic method to fundamental gap problems on the sphere

  Transactions of the American Mathematical Society · 2024-10-04

  article

  We provide a probabilistic proof of the fundamental gap estimate for Schrödinger operators in convex domains on the sphere, which extends the probabilistic proof of F. Gong, H. Li, and D. Luo [Potential Anal. 44 (2016), pp. 423–442] for the Euclidean case. Our results further generalize the results achieved for the Laplacian by S. Seto, L. Wang, and G. Wei [J. Differential Geom. 112 (2019), pp. 347–389], as well as by C. He, G. Wei, and Qi S. Zhang [Amer. J. Math. 142 (2020), pp. 1161–1191]. The essential ingredient in our analysis is the reflection coupling method on Riemannian manifolds.

  PublisherDOI

• Singular metrics with nonnegative scalar curvature and RCD

  arXiv (Cornell University) · 2024-12-12

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

• Volume entropy and rigidity for RCD-spaces

  arXiv (Cornell University) · 2024-11-07 · 1 citations

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

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

  arXiv (Cornell University) · 2024-05-12

  preprintOpen accessSenior author

  In this paper we prove that the space $\cM(n,\rv,D,Λ):=\{(M^n,g) \text{ closed }: ~~\Ric\ge -(n-1),~\Vol(M)\ge \rv&gt;0, \diam(M)\le D \text{ and } \int_{M}|\Rm|^{n/2}\le Λ\}$ 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.

  PublisherOA PDFDOI

Recent grants

• Comparison Geometry and Rigidity

  NSF · $172k · 2018–2021

• Spaces with Curvature Bounded from Below

  NSF · $151k · 2015–2019

• Problems Related to Ricci Curvature

  NSF · $108k · 2005–2008

• Manifolds with Lower Ricci Curvature Bounds

  NSF · $145k · 2008–2012

• Aspects of Bakry-Emery Ricci Curvature

  NSF · $209k · 2011–2015

Frequent coauthors

• Xianzhe Dai

  27 shared

• Christina Sormani

  20 shared

• Xuan Hien Nguyen

  Iowa State University

  10 shared

• Jiayin Pan

  9 shared

• Alina Stancu

  Concordia University

  9 shared

• Ilesanmi Adeboye

  Wesleyan University

  8 shared

• Valentina‐Mira Wheeler

  7 shared

• Rugang Ye

  7 shared

Labs

• Guofang Wei's LabPI

  Research in differential geometry and geometric analysis, focusing on global Riemannian geometry, including the study of fundamental groups, comparison geometry, and manifolds with integral curvature bounds.