arXiv subjects

Stefano Gioffrè

Publications and source records attributed to Stefano Gioffrè.

4 recordsLinked to original sources

arXiv2021

Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces

We prove that, for every closed (not necessarily convex) hypersurface $Σ$ in $\mathbb{R}^{n+1}$ and every $p>n$, the $L^p$-norm of the trace-free part of the anisotropic second fundamental form controls from above the $W^{2,p}$-closeness of $Σ$ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime $p\leq n$, the lack of convexity assumptions may lead in general to bubbling phenomena. Moreover, we obtain a stability theorem for quasi Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting.

math.AP↗

arXiv2017

Quantitative stability for anisotropic nearly umbilical hypersurfaces

We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $n \geq 2$, $p\in (1, \, +\infty)$ and $Σ$ an $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$, boundary of a convex, open set. We show that if the $L^p$ norm of the trace-free part of the anisotropic second fundamental form is small, then $Σ$ must be $W^{2, \, p}$-close to the Wulff shape, with a quantitative estimate.

math.DG↗

arXiv2017

Quantitative $W^{2, \, p}$-stability for almost Einstein hypersurfaces

It is a well known fact that, if $Σ$ is an Einstein hypersurface with positive scalar curvature, then it is a round sphere. We give a stable version of this result showing that if a hypersurface is almost Einstein in a $L^p$-sense, then it is $W^{2, \, p}$ - close to the round sphere. The result is given in a quantitative way.

math.DG↗

arXiv2017

A $W^{2, \, p}$-estimate for nearly umbilical hypersurfaces

Let $n \ge 2$, $p \in (1, \, +\infty)$ be given and let $Σ$ be a $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$. Denote by $A$ its second fundamental form, and by $\mathring{A}$ the tensor $A - \frac{1}{n} A^i_i g$ where $g = δ|_Σ$.Assuming that $Σ$ is the boundary of a convex, open set we prove that if the $L^p$-norm of $\mathring{A}$ is small, then $Σ$ must be $W^{2, \, p}$-close to a sphere, with a quantitative estimate.

math.DG↗