In this paper, we establish a general volume estimate for complete Riemannian manifolds under suitable differential inequalities involving a proper function and a symmetric tensor, without imposing curvature assumptions. As an application, we prove that a class of orientable hypersurfaces properly immersed in Euclidean space has at most polynomial volume growth and finite weighted volume. This class includes, in particular, certain self-shrinkers of the $Q_k$-flow and properly immersed minimal hypersurfaces for which the normal component of the position vector is bounded.
For $n\leq 18$, we prove that any smooth immersion of the $n$-torus into the closed unit ball in $\mathbb R^q$ has a point at which the spherical average of $\lvert II(v,v)\rvert^2$ is at least $3n/(n+2)$. This answers a question of Petrunin in these dimensions. The proof combines the scalar curvature obstruction for t...
We prove the linear stability of an embedded mean curvature flow shrinker in $\mathbb R^4$ with the topology of the M\"{o}bius bundle. This provides a non-flat, non-spherical stable shrinker in higher codimension, in sharp contrast with the classification of stable hypersurface shrinkers. Topologically, the M\"obius sh...
Kahnrad Braxton, Tang-Kai Lee, Jonathan J. Zhu· 0 citations
By constructing explicit entire maps, we prove that every exceptional Morimoto--Nagano model $M_t^3$, $t>1$, can be realized as a compact real-analytic hypersurface in $\mathbb C^3$. This extends Isaev's realization results from restricted parameter ranges to all $t>1$ and provides a complete solution to a long-standin...
We consider compact immersed surfaces of genus zero in a three-dimensional smooth metric measure space \((M^3,g,e^{-f}\mathrm{d}V_{g})\). We introduce a new weighted mean curvature and prove that any such surface whose weighted mean curvature is constant must be totally umbilical, provided that a condition relating the...
Let $(M^n,g)$ be a complete Riemannian manifold with $\Ric\ge-kg$ and uniformly positive $m$-intermediate curvature in the sense of Brendle--Hirsch--Johne. We prove that the Fisher eigenvalue $\lambda_{n-m+1}$ is small, after heat averaging, below the curvature scale $k^{-1}$. Consequently, balls have polynomial volume...
We prove that in the anti-Gaussian space $\bigl(\R^{m+1},\bar g_{\Euc},e^{|x|^2/4}\,dx\bigr)$, $m\ge2$, round spheres centered at the origin are the only closed, connected, two-sided immersed hypersurfaces with constant weighted mean curvature that are stable under weighted-volume-preserving variations.
Jinchuan Bai, Chao Xia· 0 citations
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