Let $(M^n, g, f)$ be a complete shrinking gradient Ricci soliton with constant scalar curvature. Under the assumptions that (i) $Ric \geq \frac{\nabla_{\nabla f}Ric}{f}$ on $M\setminus D$, where $D$ is a compact set over $M$; (ii) $(M^n, g, f)$ smoothly converges to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$, we conclude that $(M^n, g, f)$ is isometric to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$. Notably, condition \textup{(i)} is weaker than the radial flatness condition in \cite{Petersen-Wylie2}.
Let $(M^{2m},g,f,J)$ be a complete gradient K\"ahler--Ricci shrinker satisfying \[ \operatorname{Ric}+\nabla^2 f=\frac12 g. \] We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer $k\in\{0,\ldots,m\}$ such that \[ R\equiv k \] and \[ (M^{2m},g,J)\cong \bi...
Wen-Qi Li, Jian-Yu Ou, Guo-Qiang Wu et al.· 0 citations
Let $M$ be an $n$-dimensional complete connected self-shrinker in $\mathbb{R}^{n+p}$. Denote by $\operatorname{Ric}$ and $\mathbf{H}$ the Ricci curvature tensor and the mean curvature vector of $M$, respectively. We prove that if the self-shrinker $M$ satisfies $\operatorname{Ric}\ge(\frac{n-2}{n^2}+\varepsilon_n)|\mat...
Nirenberg's theorem gives interior $C^{2,\alpha}$ regularity for continuous viscosity solutions of fully nonlinear uniformly elliptic equations $F(D^2u)=0$ in dimension two. In dimensions five and higher, the work of Nadirashvili--Tkachev--Vl\u{a}du\c{t} shows that such solutions need not be $C^2$. We construct a famil...
We give counterexamples to Yau's question on normalized scalar curvature integrals in every dimension $n\geq4$. For each such $n$, there exists a smooth complete Riemannian metric $g$ on $\mathbb{R}^n$ with nonnegative Ricci curvature and a pole such that \[ \lim_{R\to\infty}R^{2-n}\int_{B_q(R)}\mathrm{Scal}_g\,\mathrm...
Let $(M^{n+1},g)$ be a compact Riemannian manifold with boundary. Under the assumptions $\Ric_g\geq ng$ and $\II_g\geq0$, Wang proposed a sharp strengthening of the Choi--Wang--Reilly estimate, asserting that the first nonzero Laplace eigenvalue of the boundary is at least $n$; see [J. Geom. Anal. 31 (2021)]. We dispro...
Let $(M^n,g)$ be a compact Riemannian manifold of dimension $n\ge 2$, possibly with nonempty convex $C^2$ boundary. We study the first nonzero eigenvalue $\lambda_1(M)$ of the Laplace--Beltrami operator, subject to the Neumann boundary condition when $\partial M\neq\emptyset$. Given a reference scalar curvature $R_E>0$...
Shun-Zi Guo· 0 citations
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