arXiv research

The paper proves that certain stationary hypersurfaces in high dimensions are essentially flat.

Unified framework for photon and massive particle hypersurfaces in stationary spacetimes.

We extend the results of Hardt and Simon on area-minimizing cones to prove that isolated singularities of stationary one-sided area-minimizing hypersurfaces can be locally perturbed away on the side that they are minimizing.

We obtain an explicit parametrization of stationary discs glued to some Levi non-degenerate hypersurfaces. These discs form a family which is invariant under the action of biholomorphisms. We use this parametrization to construct a local circular representation of these hypersurfaces. As a corollary, we get the uniquen…

In this paper we study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending to these spaces a result of J. Simons on the minimality of cones in Euclidean space, and apply…

In this paper, we study stability and instability problem for type-II partitioning problem. First, we make a complete classification of stable type-II stationary hypersurfaces in a ball in a space form as totally geodesic $n$-balls. Second, for general ambient spaces and convex domains, we give some topological restric…

In this paper we study the r-stability of closed spacelike hypersurfaces with constant $r$-th mean curvature in conformally stationary spacetimes of constant sectional curvature. In this setting, we obtain a characterization of $r-$stability through the analysis of the first eigenvalue of an operator naturally attached…

The paper classifies ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.

Let $N$ be a Riemannian manifold and consider a stationary union of three or more $C^{1,μ}$ hypersurfaces-with-boundary $M_k$ in $N$ with a common boundary $Γ$. We show that if $N$ is smooth, then $Γ$ is smooth and each $M_k$ is smooth up to $Γ$ (real analytic in the case $N$ is real analytic). Consequently we strength…

In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients of the metric, etc. We prove that these technical assumptions admit a natural i…

Study of spacelike submanifolds in spherical RW spacetime, proving a Lorentzian Takahashi theorem.

The Sagnac effect is re-examined using Finslerian geometry.

We study the motion of an $n$-dimensional closed spacelike hypersurface in a Lorentzian manifold in the direction of its past directed normal vector, where the speed equals a positive power $p$ of the mean curvature. We prove that for any $p\in (0,1]$, the flow exists for all time when the Ricci tensor of the ambient s…

We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local and global integral estimates. First we show that a properly immersed stationary (ΔH \equiv 0) hypersurface in \R^3 or \R^4 with restricted growth of the curvature at infinity and small total tracefree c…

The paper explores geometric invariants of null hypersurfaces using Carrollian geometry.

In this paper we study sets in the $n$-dimensional Heisenberg group $\hhn$ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in $\hhn$. We define a notion of mean curvature for hypersurfaces and we show that the boundary of a…

By using Stationary-to-Randers correspondence (SRC), a characterization of light and time-convexity of the boundary of a region of a standard stationary (n+1)-spacetime is obtained, in terms of the convexity of the boundary of a domain in a Finsler n or (n+1)-space of Randers type. The latter convexity is analyzed in d…

The paper proves stability of certain graph types in Euclidean space with specific densities.

New general results of non-existence and rigidity of spacelike submanifolds immersed in a spacetime, whose mean curvature is a time-oriented causal vector field, are given. These results hold for a wide class of spacetimes which includes globally hyperbolic, stationary, conformally stationary and pp-wave spacetimes, am…

The distance of an almost constant mean curvature boundary from a finite family of disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillation of the scalar mean curvature. This result allows one to quantitatively describe the geometry of volume-constrained stationary sets in capill…

Generalizing Riemannian theorems of Anderson-Herzlich and Biquard, we show that two $(n+1)$-dimensional stationary vacuum space-times (possibly with cosmological constant $Λ\in \R$) that coincide up to order one along a timelike hypersurface $\mycal T$ are isometric in a neighbourhood of $\mycal T$. We further prove th…

The paper proves the existence of a horizon for certain spacetimes.

Study shows persistence of singularities in minimal hypersurfaces is rare.

We prove that there exist solutions for a non-parametric capillary problem in a wide class of Riemannian manifolds endowed with a Killing vector field. In other terms, we prove the existence of Killing graphs with prescribed mean curvature and prescribed contact angle along its boundary. These results may be useful for…

In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In the author's thesis the stability of cylinders under the flows was considered; it was found that they are stable provi…

Study properties of hyperbolic Yamabe solitons in submanifolds.

For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanif…

Develops a PDE approach to constructing nontrivial anisotropic surfaces.

Study bounds singular set of minimal hypersurfaces with index control.

CMS formulation solves Poincare conjecture for all dimensions.

Hardt-Simon proved that every area-minimizing hypercone $\mathbf{C}$ having only an isolated singularity fits into a foliation of $\mathbb{R}^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $\mathbf{C}$. In this paper we prove that if a stationary $n$-varifold $M$ in the unit ball $B_1 \subset \mathbb{R}^…

New moving plane method for varifolds promotes smoothness from boundary to interior.

We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M=\R\times S$ and Randers metrics on $S$. In particular, for stationary spacetimes, we give a simple characterization of when they are causally conti…

The paper studies critical sections of the Allen-Cahn functional and their relation to minimal hypersurfaces.

We study time-like hypersurfaces with vanishing mean curvature in the (3+1) dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy problem. This solution is linearly unstable, and we show that this …

We consider an entire graph $S$ in $\mathbb R^{N+1}$ of a continuous real function $f$ over $\mathbb R^{N}$ with $N\ge 1$. Let $Ω$ be an unbounded domain in $\mathbb R^{N+1}$ with boundary $S$. Consider nonlinear diffusion equations of the form $\partial_t U= Δφ(U)$ containing the heat equation. Let $U$ be the solution…

The paper studies mean curvature flow with contact angles in high-dimensional cylinders.

Recently, a link between Lorentzian and Finslerian Geometries has been carried out, leading to the notion of wind Riemannian structure (WRS), a generalization of Finslerian Randers metrics. Here, we further develop this notion and its applications to spacetimes, by introducing some characterizations and criteria for th…

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…

Study area minimizing currents in Riemannian manifolds, proving unique structure and decay.

We give a necessary and sufficient geometric structural condition for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities; when this condition is satisfied, the singular set is empty if the…

We consider the reduced Allen-Cahn action functional, which appears as the sharp interface limit of the Allen-Cahn action functional and can be understood as a formal action functional for a stochastically perturbed mean curvature flow. For suitable evolutions of generalized hypersurfaces this functional consists of th…

New definition resolves ambiguity in non-stationary bandit classification.

The paper establishes structure theory for stable varifolds with applications to area minimising hypersurfaces.

We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such a…

It is shown that the Kerr-Newman solution, representing charged and rotating stationary black holes, admits analytic extension at the singularity. This extension is obtained by using new coordinates, in which the metric tensor becomes smooth on the singularity ring. On the singularity, the metric is degenerale - its de…

The method approximates stationary distributions of Markov models by truncating irrelevant states.

Proposes a new algorithm for non-stationary bandits.