arXiv research

A classical result of A.D. Alexandrov states that a connected compact smooth $n-$dimensional manifold without boundary, embedded in $\Bbb R^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_{n+1}=$constant in case $M$ satisfies: for any tw…

Study of Horn's problem in PU(n,1) for n≥1.

New problems on NNSC fill-ins for Bartnik data in high dimensions.

Study confirms the uniqueness of the unit sphere for a specific geometric problem.

The paper solves a curvature problem in hyperbolic space using a flow approach.

The paper proves the regularity of cohomogeneity two problems and constructs minimal hypersurfaces on spheres.

Solves Dirichlet problem for specific PSH functions on Hermitian manifolds.

The study proves stable minimal immersions in positively curved manifolds are totally geodesic.

Solves the asymptotic Plateau problem in hyperbolic space for specific curvature.

We show some characterizations of hyperspheres in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ with intrinsic and extrinsic properties such as the $n$-dimensional area of the sections cut off by hyperplanes, the $(n+1)$-dimensional volume of regions between parallel hyperplanes, and the $n$-dimensional surf…

The paper solves a geometric problem using curvature flow and variational methods.

Let $\mathbb{R}_{+}^{n+1}$ \ be the half-space model of the hyperbolic space $\mathbb{H}^{n+1}.$ It is proved that if $Γ\subset\left\{ x_{n+1}=0\right\} \subset\partial_{\infty}\mathbb{H}^{n+1}$ is a bounded $C^{0}$ Euclidean graph over $\left\{ x_{1}=0,\text{ }x_{n+1}=0\right\}$ then, given $\left\vert H\right\vert <…

We solve Blaschke's problem for hypersurfaces of dimension $n\geq 3$. Namely, we determine all pairs of Euclidean hypersurfaces $f,\tilde{f}\colon M^n\to\R^{n+1}$ that induce conformal metrics on $M^n$ and envelope a common sphere congruence in $\R^{n+1}$.

Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in $H^{n+1}$, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with t…

We define a family ${\rm KV}^{(g,n)}$ of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus $g$ with $n+1$ boundary components. The problem ${\rm KV}^{(0,3)}$ is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of ${\rm KV}^{(g,n)}$ for ar…

We consider a solution f of a certain Dirichlet Problem on a domain in $S^{(n+1)}$ whose boundary is a minimal hypersurface and we prove a Poincare type inequality for f. One have equality iff Yau's conjecture about the first non-zero eigenvalue of closed minimal hypersurfaces of $S^{(n+1)}$ is true.

Study rolling control of Lorentzian manifolds on flat space.

Inverse scattering result on AH manifolds determines metric up to diffeo and conformal factor.

Proves inequalities for hypersurfaces in the sphere, solving a long-standing problem.

We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of…

Study flows for capillary Minkowski problems in half-spaces.

Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $\int_MH dv$ of the mean curvatur…

In this paper, the pinching problems of complete $λ$-hypersurfaces in a Euclidean space $\mathbb R^{n+1}$ are studied. By making use of the Sobolev inequality, we prove a global pinching theorem of complete $λ$-hypersurfaces in a Euclidean space $\mathbb R^{n+1}$.

Let $Z \to Y^{2n+1}$ be the bundle of Legendrian $n$-planes over a contact manifold $Y$. We consider a foliation of $Z$ by canonical lifts of Legendrian submanifolds, called \emph{Legendrian submanifold path geometry}, whose flat model is \[ Sp(n+1, R) \to RP^{2n+1}. \] The equivalence problem provides an $sp(n+1, R)$ …

Paper solves Christoffel-Minkowski problem in hyperbolic space.

We present a characterisation of Maurer-Cartan 1-superforms associated to the two-dimensional supersymmetric $\mathbb{C}P^{N-1}$ sigma model. We, then, solve the associated linear spectral problem and use its solutions to describe an integrable system for a $su(N)$-valued map.

The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.

In this paper we introduce a new geometric flow --- the hyperbolic gradient flow for graphs in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$. This kind of flow is new and very natural to understand the geometry of manifolds. We particularly investigate the global existence of the evolution of convex hypers…

The geometry of canal hypersurfaces of an n-dimensional conformal space C^n is studied. Such hypersurfaces are envelopes of r-parameter families of hyperspheres, 1 \leq r \leq n-2. In the present paper the conditions that characterize canal hypersurfaces, and which were known earlier, are made more precise. The main at…

The problem of determining the {\it Bonnet hypersurfaces in} $R^{n+1}$, for $n>1$, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/o…

Symmetric hypersurfaces and boundaries in R^n+1 with group actions.

In the 1920's Marston Morse developed what is now known as Morse theory trying to study the topology of the space of closed curves on S^2. We propose to attack a very similar problem, which 80 years later remains open, about the topology of the space of closed curves on S^2 which are locally convex (i.e., without infle…

Study of constant curvature hypersurfaces in hyperbolic space.

Researchers find counterexamples to inverse problems for wave equations.

The paper proves nonexistence of NNSC cobordism for Bartnik data under certain conditions.

The paper proves uniqueness of solutions to curvature problems using various methods.

Constructs hyperspheres with prescribed mean curvature in Euclidean space.

We consider the problem of extending functions φ:\to S^n to functions u:B^{n+1}\to S^n for n=2,3. We assume φto belong to the critical space W^{1,n} and we construct a W^{1,(n+1,\infty)}-controlled extension u. The Lorentz-Sobolev space W^{1,(n+1,\infty)} is optimal for such controlled extension. Then we use such resul…

We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincaré $(2n-1)$-complex such that $H^{n-1}(X;\mathbb{Q})=0$ and $H^{n-1}(X;\mathbb{Z}_2)=0$. The…

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean curvature function. Let $X:M\to \mathbb{R}^{n+1}$ be an $n$-dimensional closed hypersu…

The paper solves curvature flow problems to prove sphere convergence and dual Minkowski solutions.

The paper explores singular minimal translation graphs in Euclidean spaces.

We study conformal symmetry breaking differential operators which map differential forms on $\mathbb{R}^n$ to differential forms on a codimension one subspace $\mathbb{R}^{n-1}$. These operators are equivariant with respect to the conformal Lie algebra of the subspace $\mathbb{R}^{n-1}$. They correspond to homomorphism…

Paper solves Hessian quotient equations in Lorentz-Minkowski space with Dirichlet boundary conditions.

Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms $Ω_{N-1}$ of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by the systems of the N-1 quasi-linear differential equations, describe coisotropic deform…

New non-quadratic Euclidean complete affine maximal type hypersurfaces found for N≥2, θ∈(0,(N-1)/N].

Estimates for stable minimal hypersurfaces in Euclidean space.

The round sphere is stable among spin manifolds with a specific scalar curvature bound.