Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.34835

Obata theorem of $σ_k$-flat and constant associated $k$th boundary curvature equation

Xuezhang Chen·Wei Wei
View original source ↗Read source document ↗Export metadata ↓

Abstract

On the upper hemisphere, we use the Obata-Escobar argument to establish an Obata theorem for $σ_k$-flat and positive constant boundary $\mathcal B_k$-curvature equation, which was initiated by Case-Wang [Adv. Math. 337 (2018), 83-106].

Explore related subjects

math.DGmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xuezhang Chen, Wei Wei. 2026-09-28. Obata theorem of $σ_k$-flat and constant associated $k$th boundary curvature equation. https://arxiv.org/abs/2609.34835

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fedosov manifolds with parallel symplectic Weyl curvature

A classical 1977 result by Roter states that a Riemannian manifold of dimension at least four with parallel Weyl curvature must be locally symmetric or conformally flat. We establish Roter's theorem in the symplectic category: namely, a Fedosov manifold with parallel symplectic Weyl curvature must be locally symmetric or Weyl-flat.

math.DG↗
arXiv2026

Steady gradient Ricci Yang-Mills solitons on 2-orbifolds

Following the recent work of M. Womack \cite{Wo26} we consider the steady Ricci Yang-Mills solitons on 2d surfaces containing orbifold point $\mathbb{R}^2/\mathbb{Z}_p$. We show that there are a family of solitons depending parameter $λ$ which approaches to $\mathbb{Z}_p$-quotient of Hamilton cigar soliton as $λ\to (-2/p)^+$ and approaches to (after rescaling) $\mathbb{Z}_p$-quotient of round sphere as $λ\to \infty$. For any integer $q >p$ there is a $λ$ whose corresponding soliton metric is defined on football orbifold $S_{p,q}^2$.

math.DG↗
arXiv2026

Green-Function Monotonicity under Positive Bakry-Émery Curvature

Inspired by the Green-function monotonicity formulas of Colding, Minicozzi, and Manea, we establish weighted monotonicity formulas on closed weighted Riemannian manifolds satisfying ${\rm Ric}_f^m \ge (m-1)kg$, where $m>n\ge 3$ and $k>0$. Our results extend Manea's elliptic monotonicity framework to the setting of finite-dimensional Bakry-Émery curvature and provide a positive-curvature counterpart to the weighted formulas of Song-Wei-Wu. Starting from the Green function of $-Δ_f+m(m-2)k/4$, we derive an exact weighted Bochner identity and prove three monotonicity formulas, together with a one-parameter family for $β\ge (m-2)/(m-1)$, under explicit pole-integrability conditions. We also treat the case of negative effective dimension $m<0$, for which the nonnegative square decomposition remains valid, while the change in the behavior near the Green pole leads to reversed area and volume monotonicity on finite sublevel sets. We conclude with comparisons and questions concerning entropy monotonicity formulas and positive-curvature RCD spaces.

math.DG↗