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Jaime B. Ripoll

Publications and source records attributed to Jaime B. Ripoll.

4 recordsLinked to original sources
arXiv2019

Asymptotic Plateau problem for prescribed mean curvature hypersurfaces

We prove the existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature in Cartan-Hadamard manifolds $N$. More precisely, given a suitable subset $L$ of the asymptotic boundary of $N$ and a suitable function $H$ on $N$, we are able to construct a set of locally finite perimeter whose boundary has generalized mean curvature $H$ provided that $N$ satisfies the so-called strict convexity condition and that its sectional curvatures are bounded from above by a negative constant. We also obtain a multiplicity result in low dimensions.

math.DG↗
arXiv2016

The Gauss map on translational Riemannian manifolds and the topology of hypersurfaces

We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this Gauss map to prove that if $M^{n}$ is a compact, connected and oriented immersed hypersurface of the unit sphere $\mathbb{S}^{n+1}$ ($n\geq2$) contained in a geodesic ball of radius $R$ and whose principal curvatures are strictly bigger than $\tan\left( R/2 \right)$, then $M$ is diffeomorphic to $\mathbb{S}^{n}$. Additionally, we show that for any $\varepsilon\in(0,\sqrt{2}-1)$ there exists a compact, connected and oriented immersed hypersurface $M_{\varepsilon}$ of $\mathbb{S}^{n+1}$ whose principal curvatures are strictly bigger than $\varepsilon \tan \left( R/2 \right)$ but $M_{\varepsilon}$ is not homeomorphic to a sphere. Finally, using this previous result, we reobtain a theorem of Qiaoling Wang and Changyu Xia (see [4]) which asserts that if a compact and oriented hypersurface of $\mathbb{S}^{n+1}$ is contained in an open hemisphere and has nowhere zero Gauss-Kronecker curvature, then it is diffeomorphic to $\mathbb{S}^n$.

math.DG↗