Related papers: Stable $(r+1)$-th capillary hypersurfaces
We study the space of smooth marked hypersurfaces in a given linear system. Specifically, we prove a homology h-principle to compare it with a space of sections of an appropriate jet bundle. Using rational models, we compute its rational…
We show that surface solitons in the one-dimensional nonlinear Schr\"odinger equation with truncated complex periodic potential can be stabilized by linear homogeneous losses, which are necessary to balance gain in the near-surface channel…
We develop a global theory for complete hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces in…
In this note we derive an upper bound for the Hausdorff dimension of the stable set of a hyperbolic set $\Lambda$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold. As a consequence we obtain that $\dim_H W^s(\Lambda)=n$ is…
The relationship between stable holomorphic vector bundles on a compact complex surface and the same such objects on a blowup of the surface is investigated, where "stability" is with respect to a Gauduchon metric on the surface and…
In this paper, we study the prescribed $k$-th Weingarten curvature problem for convex capillary hypersurfaces in $\overline{\mathbb{R}^{n+1}_+}$. This problem naturally extends the prescribed $k$-th Weingarten curvature problem for closed…
Recently ({\em Class. Quant. Grav.} {\bf 20} 625-664) the concept of {\em causal mapping} between spacetimes --essentially equivalent in this context to the {\em chronological map} one in abstract chronological spaces--, and the related…
Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width $D$.…
We generalize the stable graph regularity lemma of Malliaris and Shelah to the case of finite structures in finite relational languages, e.g., finite hypergraphs. We show that under the model-theoretic assumption of stability, such a…
We study stationary capillary-gravity waves in a two-dimensional body of water that rests above a flat ocean bed and below vacuum. This system is described by the Euler equations with a free surface. Our main result states that there exist…
We consider convex, spacelike hypersurfaces with boundaries on some hyperboloid (or lightcone) in the Minkowski space. If the hypersurface has constant higher order mean curvature, and the angle between the normal vectors of the…
A hypersurface $M$ in ${\bf R}^n$ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if the number of distinct principal curvatures is…
We show that if $M^n$ is a properly immersed, two-sided, stable minimal hypersurface in $B^{n+1}_1(0)\setminus S$, where $S$ is closed with $\mathcal{H}^{n-2}(S)=0$, then $\text{dim}_{\mathcal{H}}\text{sing}(M)\leq n-7$, namely…
We consider hypersurfaces of products $M\times\mathbb R$ with constant $r$-th mean curvature $H_r\ge 0$ (to be called $H_r$-hypersurfaces), where $M$ is an arbitrary Riemannian $n$-manifold. We develop a general method for constructing…
We define a natural notion of higher order stability and show that subsets of $\mathbb{F}_p^n$ that are tame in this sense can be approximately described by a union of low-complexity quadratic varieties, up to linear error. This generalizes…
In this paper, we consider capillary-gravity waves propagating on the interface separating two fluids of finite depth and constant density. The flow in each layer is assumed to be incompressible and of constant vorticity. We prove the…
This paper establishes the conditional orbital stability of fully localized solitary waves for the three-dimensional capillary-gravity water wave problem in finite depth under strong surface tension. The waves, constructed via a…
Planes and circular cylinders are models of interfaces of a fluid when the support surface is translationally invariant in a direction of the space. After a study of the eigenvalues of the Jacobi operator, it is investigated when planar…
In this note, we prove that for every $0<\sigma<1$, there exists a smooth complete hypersurface $\Sigma$ in $\mathbb{H}^{n+1}$ with prescribed asymptotic boundary $\partial \Sigma=\Gamma$ at infinity, whose principal curvatures…
The Cauchy problem for the parabolic--elliptic Keller--Segel system in the whole $n$-dimensional space is studied. For this model, every constant $A \in \mathbb{R}$ is a stationary solution. The main goal of this work is to show that $A <…