arXiv research

Boundary of fiber convex domains is a cohomological sphere.

The study proves convergence of conic 4-spheres' geometry to boundary cases.

Hyperbolic groups act on spheres, and nearby actions are semi-conjugate.

The paper studies rigidity of sphere packings on 3D manifolds with boundary.

The paper studies rigid sphere packings on 3D manifolds with boundary.

Proves condition for 4-manifolds with sphere boundary to be standard.

Free boundary minimal submanifolds with boundaries on concentric spheres

Spheres in curve graphs are connected, proving Gromov boundary linearity.

Minimal surfaces reflect across spheres, proving annulus uniqueness.

Study on nonlinear equations on spheres and hemispheres with zero Neumann boundary condition.

Existence of minimal annuli in 3-sphere with boundary on geodesic spheres.

We use the inverse mean curvature flow with a free boundary perpendicular to the sphere to prove a geometric inequality involving the Willmore energy for convex hypersurfaces of dimension $n\geq 3$ with boundary on the sphere.

In this paper we construct Ricci-positive metrics on the connected sum of products of arbitrarily many spheres provided the dimensions of all but one sphere in each summand are at least 3. There are two new technical theorems required to extend previous results on sums of products of two spheres. The first theorem is a…

New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.

Research shows conditional existence of foliations by CMC and Willmore type half-spheres near a boundary point.

Detects exotic embeddings in 4-manifolds with boundary.

New 3D handlebodies in 4-sphere and 5-ball are not isotopic even with same boundary.

In this paper we prove that the closed $4$-ball admits non-Kähler complex structures with strictly pseudoconcave boundary. Moreover, the induced contact structure on the boundary $3$-sphere is overtwisted.

A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and tra…

We present a direct proof that the Anomaly Boundary term of J. Brünning and X. Ma generalizes to the cases of the cone over a $m$-dimensional sphere.

Minimal triangulations of spheres map almost linearly to boundaries of high-dimensional polytopes.

We describe, under some additional technical assumptions, the Gromov boundary of the free product of several $G_i$'s amalgamated wrt. $H$, where $G_i$ are hyperbolic groups with boundary homeomorphic to a densely punctured $n$-sphere, and $H$ is their common subgroup corresponding to a peripheral sphere in each of the …

The paper constructs four-manifolds with lens space boundaries and explores sphere configurations in $\#^n \mathbb{C} P^2$.

We discuss the concept of the shadow boundary of a centrally symmetric convex ball $K$ (actually being the unit ball of a Minkowski normed space) with respect to a direction ${\bf x}$ of the Euclidean n-space $R^n$. We introduce the concept of general parameter spheres of $K$ corresponding to this direction and prove t…

We introduce the non-pure versions of simplicial balls and spheres with minimum number of vertices. These are a special type of non-homogeneous balls and spheres (NH-balls and NH-spheres) satisfying a minimality condition on the number of maximal simplices. The main result is that minimal NH-balls and NH-spheres are pr…

Paper solves local well-posedness for Schrödinger flow into sphere with natural boundary conditions.

Study on holomorphic curves in 6-sphere with boundary conditions.

Let G be a torsion-free hyperbolic group and let n > 5 be an integer. We prove that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n-1)-dimensional sphere.

Inverse mean curvature flow converges to a disk in hyperbolic space.

Paper solves a conjecture about minimal surfaces using sphere intersections and Weierstrass data.

The paper solves a Nirenberg problem on half spheres, finding multiple blow-ups.

We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.

We extend to higher dimensions earlier sharp bounds for the area of two dimensional free boundary minimal surfaces contained in a geodesic ball of the round sphere. This follows work of Brendle and Fraser-Schoen in the euclidean case.

Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle. We study under what geometric conditions the surface must …

Constructs minimal surfaces near the boundary of a ball.

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.

Proves convergence groups on a 2-sphere are Kleinian groups.

Derives an inequality for submanifolds in spheres.

New solutions found for a complex boundary problem.

The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon's Conjecture: A hyperbolic group $G$ (that acts effectively…

We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower bound $n(n-1)$ is critical in the sense that the hypersurface may contain geode…

Ozsváth and Szabó used the knot filtration on $\widehat{CF}(S^3)$ to define the $τ$-invariant for knots in the 3-sphere. In this article, we generalize their construction and define a collection of $τ$-invariants associated to a knot $K$ in a rational homology sphere $Y$. We then show that some of these invariants prov…

Existence proved for static vacuum extensions near Schwarzschild spheres.

In this paper, we characterize non-hyperbolic 3-component links in the 3-sphere whose exteriors contain essential 3-punctured spheres with non-integral boundary slopes. We also show the existence of embeddings of some multibranched surfaces in the 3-sphere which satisfy some homological conditions to be embedded in the…

We prove that in Euclidean space $R^{n+1}$ any compact immersed nonnegatively curved hypersurface $M$ with free boundary on the sphere $S^n$ is an embedded convex topological disk. In particular, when the $m^{th}$ mean curvature of $M$ is constant, for any $1\leq m\leq n$, $M$ is a spherical cap or an equatorial disk.

The study finds the optimal metrics for free boundary minimal surfaces in spherical caps.

Maximizes arcs on a sphere with constraints.