arXiv subjects

Benoît Daniel

Publications and source records attributed to Benoît Daniel.

4 recordsLinked to original sources

arXiv2024

Uniform estimates for a family of Poisson problems: `rounding off the corners'

We prove \emph{uniform solvability estimates} for certain families of elliptic problems posed in a bounded family of domains (for example, a sequence that converges to another domain). We provide uniform estimates both in weighted and in usual Sobolev spaces. When the limit domain is a \emph{polygon} and the other domains are smooth, our results amount to rounding off'' the corners of the limit domain. The technique of proof is based on a suitable conformal modification of the metric, which makes the union of the domains a manifold with boundary and relative bounded geometry.

math.AP↗

arXiv2023

Generalized Ricci surfaces

We consider smooth Riemannian surfaces whose curvature $K$ satisfies the relation $Δ\log|K-c|=aK+b$ away from points where $K=c$ for some $(a,b,c)\in\mathbb{R}^3$, which we call generalized Ricci surfaces. We prove some isometric immersion theorems allowing points where $K=c$ using properties of log-harmonic functions. For instance, we obtain a characterization of Riemannian surfaces that locally admit minimal isometric immersions, possibly with umbilical points, into a $3$-dimensional Riemannian manifold of constant sectional curvature. We also give an application to convex affine spheres. Finally, we study compact generalized Ricci surfaces: we obtain topological obstructions and construct examples.

math.DG↗

arXiv2019

Constant mean curvature Isometric Immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and related results

In this article, we study constant mean curvature isometric immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and we classify these isometric immersions when the surface has constant intrinsic curvature. As applications, we use the sister surface correspondence to classify the constant mean curvature surfaces with constant intrinsic curvature in the $3-$dimensional homogenous manifolds $\mathbb{E}(κ, τ)$ and we use the Torralbo-Urbano correspondence to classify the parallel mean curvature surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{H}^2 \times \mathbb{H}^2$ with constant intrinsic curvature. It is worthwhile to point out that these classifications provide new examples.

math.DG↗

arXiv2010

The Gauss map of minimal surfaces in the Heisenberg group

We study the Gauss map of minimal surfaces in the Heisenberg group $\mathrm{Nil}_3$ endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane $\mathbb{H}^2$. Conversely, any nowhere antiholomorphic harmonic map into $\mathbb{H}^2$ is the Gauss map of a nowhere vertical minimal surface. Finally, we study the image of the Gauss map of complete nowhere vertical minimal surfaces.

math.DG↗