arXiv research

Sharp convergence theorem for sphere submanifolds proved.

Spheres' spectral structure converges to Gaussian space's as dimensions grow.

We prove a convergence theorem on the moduli space of constant $σ_{2}$ metrics for conic 4-spheres. We show that when a numerical condition is convergent to the boundary case, the geometry of conic 4-spheres converges to the boundary case while preserving capacity.

In this paper, we investigate Liu-Xu-Ye-Zhao's conjecture [30] and prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in spheres which improves the convergence theorem of Baker [2] as well as the differentiable sphere theorems of Gu-Xu-Zhao [16, 50, 52].

We show that spheres of positive constant curvature with $n$ ($n\geq3$) conic points converge to a sphere of positive constant curvature with two conic points (or called an (American) football) in Gromov-Hausdorff topology when the corresponding singular divisors converge to a critical divisor in the sense of Troyanov.…

This study shows the moment-SOS hierarchy converges in polynomial optimization over product of spheres.

Study on sphere-valued maps, proving energy convergence and current limits.

Survey on mean curvature flow with sphere theorems and Yau rigidity theory.

CR Yamabe flow fails to converge on small deformations of the standard CR three-sphere.

Uniform convergence of metrics on vortex moduli space in Bradlow limit.

Random harmonic maps into spheres converge to a specific metric under strong convergence of representations.

We prove that if the initial hypersurface of the mean curvature flow in spheres satisfies a sharp pinching condition, then the solution of the flow converges to a round point or a totally geodesic sphere. Our result improves the famous convergence theorem due to Huisken [9]. Moreover, we prove a convergence theorem und…

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

Flow preserves quermassintegrals, converging to a geodesic sphere.

Study on Ricci flow on 4-spheres, proving standard sphere convergence.

Study on stability of mean curvature flow in hyperbolic space.

The paper studies curvature flows of star-shaped hypersurfaces and proves convergence to spheres.

In a recent paper Donaldson defines three operators on a space of Hermitian metrics on a complex projective manifold: $T, T_ν, T_K.$ Iterations of these operators converge to balanced metrics, and these themselves approximate constant scalar curvature metrics. In this paper we investigate the convergence properties of …

The paper studies how submanifolds of a sphere evolve over time.

The paper studies curvature flows in hyperbolic space and proves convergence to spheres under certain conditions.

Random hyperbolic surfaces with punctures converge to the Brownian sphere.

The paper studies non-integer curvature flows and proves convergence to spheres under specific conditions.

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

New surfaces near a sphere violate Minkowski inequality.

The paper proves global existence and convergence of Möbius-invariant Willmore flow in 3-sphere.

The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is given. The semi-stable and unstable cases are new, and it is shown that the flow …

Study on prescribing positive curvature with conical singularities on a sphere.

Given a 3-dimensional Riemannian manifold $(M,g)$, we prove that if $(Φ_k)$ is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres), having Willmore energy bounded above uniformly strictly by $8 π$, and Hausdorff converging to a point $\bar{p}\in M$, then $Scal(\bar{p})=0$ and $\nabla Sc…

This is the first of two papers, in which we prove some properties of the Webster scalar curvature flow. More precisely, we establish the long-time existence, L^p convergence and the blow-up analysis for the solution of the flow. As a by-product, we prove the convergence of the CR Yamabe flow on the CR sphere. The resu…

The Willmore flow preserves surface volume, leading to convergence to a sphere.

In this paper, we prove some convergence theorems for the mean curvature flow of closed submanifolds in the unit sphere $\mathbb{S}^{n+d}$ under integral curvature conditions. As a consequence, we obtain several differentiable sphere theorems for certain submanifolds in $\mathbb{S}^{n+d}$.

We show that on a Sasakian 3-sphere the Sasaki-Ricci flow initiating from a Sasakian metric of positive transverse scalar curvature converges to a gradient Sasaki- Ricci soliton. We also show the existence and uniqueness of gradient Sasaki-Ricci soliton on each Sasakian 3-sphere.

The paper studies a modified scalar curvature flow and proves convergence to a sphere.

Study approximates product of spheres using Laplacian eigenvalues.

We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in $\mathbb{C}^n$ satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.

We construct hyperbolic integer homology 3-spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3-manifolds which Benjamini-Schramm converge to H^3 whose normalized Ray-Singer analytic torsions do not converge to …

New convexity concept applied to sphere yields quermassintegral inequalities.

Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is homotopic to a constant…

Study finds many nonplanar minimal spheres in elongated ellipsoids.

We prove: "If $M$ is a compact hypersurface of the hyperbolic space, convex by horospheres and evolving by the volume preserving mean curvature flow, then it flows for all time, convexity by horospheres is preserved and the flow converges, exponentially, to a geodesic sphere". In addition, we show that the same conclus…

Study shows limits of metrics with positive scalar curvature on spheres.

We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences we show that there exist examples where eve…

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…

The paper examines conditions for Gromov-Hausdorff convergence of metric quotients and provides examples of conic-flat surfaces.

The paper proves geometric inequalities in sphere using locally constrained flows.

New optimization model converges to global minimizers on spheres.

We consider contracting and expanding curvature flows in $\Ss$. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gauß map. The contracting hypersurfaces shrink to a point $x_0$ while the expanding hypersur…

We show that strictly convex surfaces expanding by the inverse Gauss curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.