arXiv research

Let $T\subset \R^{m+1}$ be a strictly convex domain bounded by a smooth hypersurface $X=\partial T$. In this paper we find lower bounds on the number of billiard trajectories in $T$ which have a prescribed intial point $A\in X$, a prescribed final point $B\in X$ and make a prescribed number $n$ of reflections at the bo…

Study finds solutions to Yamabe equation with specific behavior near singular points.

Study on polygons with fixed edge slopes and their perimeter function.

Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…

Study on scalar curvature minimizability loss and saddle point solutions.

Proves existence of sphere foliations with prescribed mean curvature on Riemannian manifolds.

We show that every source connected Lie groupoid always has global bisections through any given point. This bisection can be chosen to be the multiplication of some exponentials as close as possible to a prescribed curve. The existence of bisections through more than one prescribed points is also discussed. We give som…

Constructs hyperspheres with prescribed mean curvature in Euclidean space.

Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.

Study variational properties of curves in half-plane with area constraints.

In this note, we study Q-curvature flow on $S^4$ with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on $S^4$ has a solution provided the prescribed Q-curvature $f$ has its positive part, which possesses non-degenerate critical points such that $Δ_{S^4} f\not=0$ at the saddle points and …

Study shows equivalence of two methods for solving scalar curvature problem.

Researchers examine global properties of a scalar curvature functional to solve the prescribed Ricci curvature problem.

Extends Y.Eliashberg's h-principle to generic maps with prescribed Thom-Boardman singularities.

Sharp area estimates for minimal submanifolds in curved spaces.

We prove the existence and uniqueness of radial graphs over a given domain of $\mathbb{S}^{n}$ having boundary on the sphere $\mathbb{S}^{n}$ and whose mean curvature at every point equals a prescribed positive function satisfying suitable barrier-type and monotonicity conditions.

This paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.

Minimal surfaces in a ball have a minimum area bound.

Study Palais-Smale sequences for Ricci curvature on homogeneous spaces.

For any prescribed closed subset of a line segment in Euclidean 3-space, we construct a sequence of minimal disks that are properly embedded in an open solid cylinder around the segment and that have curvatures blowing up precisely at the points of the closed set.

Paper solves curvature assignment on surfaces with sharp points and edges.

New existence results for curvature problem on balls with specific conditions.

The study finds solutions to a curvature minimisation problem in fixed-length curves.

The classical $k$-means algorithm for partitioning $n$ points in $\mathbb{R}^d$ into $k$ clusters is one of the most popular and widely spread clustering methods. The need to respect prescribed lower bounds on the cluster sizes has been observed in many scientific and business applications. In this paper, we present an…

We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic $p \geq 2$. In this cas…

Reproves results on spherical metrics using parabolic bundles.

Non-compact flow lines for scalar curvature prescription on manifolds.

New real algebraic maps with prescribed images and compositions are constructed locally like moment maps.

We construct a sequence of compact embedded minimal disks in a ball in Euclidean 3-space, whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on the x_3-axis. This extends a result of Colding and Minicozzi, who constructed a …

In this paper we consider surfaces of class $C^1$ with continuous prescribed mean curvature in a three-dimensional contact sub-Riemannian manifold and prove that their characteristic curves are of class $C^2$. This regularity result also holds for critical points of the sub-Riemannian perimeter under a volume constrain…

Choose two points in the tangent bundle of the Euclidean plane $(x,X),(y,Y)\in T{ \mathbb R}^2$. In this work we characterise the immersed length minimising paths with a prescribed bound on the curvature starting at $x$, tangent to $X$; finishing at $y$, tangent to $Y$, in each connected component of the space of paths…

Quaternionic reformulation simplifies surface curvature theory.

Existence and uniqueness in ${\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\overrightarrow{Om}$ pointing in the direction of $\overrighta…

Globally hyperbolic spacetimes admitting infinitely many causal (and timelike) homotopy classes of curves joining two prescribed points, are exhibited and discussed.

We consider a $C^{1}$ smooth surface with prescribed $p$(or $H$)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed $p$-mean curvature $H\in C^{0},$ we show that any characteristic curve is $C^{2}$ smooth and its (line) curvature equals $-H$ in the nonsingular domain$.$ By introducing ch…

Noninjective monodromy found in polynomial critical point tracking.

New method constructs smooth functions with specific Reeb graphs and preimages on 3D manifolds.

Study reconstructs Morse-Bott functions with specific preimage conditions on 3D manifolds.

The paper studies metrics that match prescribed geodesics and introduces a variational problem.

In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface Σin a compact n-manifold M, there is a Riemannian metric on M such that the nodal set of its first nontrivial eigenfunction is Σ. We present…

We study the problem of prescribing the Paneitz curvature on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of suc…

Constructs surfaces with specific topologies and curvatures.

Global existence and convergence proved for Kazdan-Warner equation with non-negative prescribed function.

Finite number of metrics found for specific spherical angles.

Researchers reconstruct algebraic maps onto curves based on prescribed Reeb graphs.

Introduces electrostatic manifolds with boundary for curvature problems.

Smooth approximations near singularities of constant mean curvature surfaces are found.

Gradient flow solves curvature problem on surfaces with conical points.