Inspired by the Simon conjecture for minimal surfaces in spheres, we study closed surfaces with parallel mean curvature vector and positive Gaussian curvature immersed in unit spheres. Let $h$ be the second fundamental form, let $\mathbf{H}$ be the mean curvature vector field, and set $\tilde h=h-\mathbf{H}g$ and $\tilde S=|\tilde h|^2=|h|^2-2H^2$, where $H=|\mathbf{H}|$ and $g$ is the induced Riemannian metric on the surface $M$. We establish three Simons-type integral identities for $\tilde S$, which extend the first, second and third gap identities in the minimal case. As applications, we obtain the first two sharp endpoint gaps and several rigidity and oscillation estimates in the third interval. We further characterize the endpoint cases by combining these identities with the classification theorems of Calabi and Yau.
We study complete
n
-dimensional submanifolds immersed in the Gaussian space
$$\mathbb {G}^{n+p},$$
that is, the Euclidean space
$$\mathbb {R}^{n+p}$$
endowed with its standard metric and the Gaussian weight
$$e^{-|x|^2/4}$$
. More precisely, in codimension
$$p\ge 2,$$
we consider submanif...
W. Barboza, H. D. de Lima, Wallace Gomes· Annali di Matematica Pura ed...· 0 citations
For immersions of a closed $2h$-dimensional manifold $\Sigma$ into $\mathbb R^m$, we consider curvature energies that are invariant under ambient isometries and dilations, with the principal term $\int_{\Sigma} \big|\nabla^{(h-1)} \vec{\mathrm{I\!I}}\big|_g^2\,d\text{vol}_g$ and lower-order submanifold invariants. Foll...
In this paper we study the global geometry and classification of complete minimal surfaces with embedded ends and finite total curvature in the product space $\mathbb{H}^2 \times \mathbb{R}$. We describe the structure of embedded ends of such surfaces, constraining the combinatorics of their asymptotic polygons at infi...
Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption an...
Let $M^n$ be a closed minimal hypersurface of the unit sphere with constant squared norm $S$ of the second fundamental form. For $n\geq4$, we prove that $S>n$ implies $S>85n/59$, improving the bound $10n/7$ of Suh and Yang. No constancy assumption is imposed on the cubic trace $f_3$. The proof combines a symmetrized He...
Let $n\geq3$ and $0\leq\tau\leq2$. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive $\tau$-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindric...
Tian-Ci Luo, Yong Wei, Rong Zhou· 0 citations
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