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Asun Jiménez

Publications and source records attributed to Asun Jiménez.

4 recordsLinked to original sources
arXiv2024

A lower bound for the first eigenvalue of a minimal hypersurface in the sphere

Let $Σ$ be a closed embedded minimal hypersurface in the unit sphere $\mathbb{S}^{m+1}$ and let $Λ=\max\limits_Σ|A|$ be the norm of its second fundamental form. In this work we prove that the first eigenvalue of the Laplacian of $Σ$ satisfies $$λ_1(Σ)> \dfrac{m}{2}+\frac{m(m+1)}{32(12Λ+m+11)^2+8},$$ and $λ_1(Σ)=m$, when $Λ\le\sqrt{m}$. In particular, this estimate improves the one obtained recently in \cite{duncan2023improved}. The proof of our main result is based on a Rayleigh quotient estimate for a harmonic extension of an eigenfunction of the Laplacian of $Σ$ in the spirit of \cite{choi1983first}.

math.DG↗
arXiv2014

Isolated singularities of graphs in warped products and Monge-Ampère equations

We study graphs of positive extrinsic curvature with a non-removable isolated singularity in 3-dimensional warped product spaces, and describe their behavior at the singularity in several natural situations. We use Monge-Ampère equations to give a classification of the surfaces in 3-dimensional space forms which are embedded around a non-removable isolated singularity and have a prescribed, real analytic, positive extrinsic curvature function at every point. Specifically, we prove that this space is in one-to-one correspondence with the space of regular, analytic, strictly convex Jordan curves in the 2-dimensional sphere $\mathbb{S^2}$.

math.DG↗
arXiv2013

A classification of isolated singularities of elliptic Monge-Ampère equations in dimension two

Let $\mathcal{M}_1$ denote the space of solutions $z(x,y)$ to an elliptic, real analytic Monge-Ampère equation ${\rm det} (D^2 z)=φ(x,y,z,Dz)>0$ whose graphs have a non-removable isolated singularity at the origin. We prove that $\mathcal{M}_1$ is in one-to-one correspondence with $\mathcal{M}_2\times Z_2$, where $\mathcal{M}_2$ is a suitable subset of the class of regular, real analytic strictly convex Jordan curves in $R^2$. We also describe the asymptotic behavior of solutions of the Monge-Ampère equation in the $C^k$-smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation ${\rm det} (D^2 z +\mathcal{A}(x,y,z,Dz))=φ(x,y,z,Dz)>0$.

math.AP↗