arXiv research

Proves optimal regularity for sphere minimizers in 3-sphere.

In this paper, we prove that the $3$-sphere endowed with an arbitrary Riemannian metric either contains at least two embedded minimal $2$-spheres or admits an optimal foliation by $2$-spheres. This generalizes recent results by Haslhofer-Ketover (Duke Math. J. 2019), where the existence of optimal foliations and minima…

New optimal surfaces found in Heisenberg group defy Euclidean sphere optimality.

Sharp bounds found on shortest geodesic on punctured spheres.

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

Study optimal partitions on spheres using fractional Q-curvature and variational methods.

Derives an inequality for submanifolds in spheres.

Optimal controls for conformal Laplacian obstacle problems on spheres and manifolds.

In this paper the regularity of optimal transportation potentials defined on round spheres is investigated. Specifically, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger and Wang is satisfied on the round sphere, when the cost-function is the geodes…

New theorem shows nearly spherical manifolds can be mapped from spheres.

We give sufficient conditions on initial and target measures supported on the sphere $§^n$ to ensure the solution to the optimal transport problem with the cost $|x-y|^2/2$ is a diffeomorphism.

The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.

Using a ramified cover of the two-sphere by the torus, we prove a local optimal inequality between the diastole and the area on the two-sphere near a singular metric. This singular metric, made of two equilateral triangles glued along their boundary, has been conjectured by E. Calabi to achieve the best ratio area over…

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

In this paper, we study the rigidity theorem of closed minimally immersed Legendrian submanifolds in the unit sphere. Utilizing the maximum principle, we obtain a new characterization of the Calabi torus in the unit sphere which is the minimal Calabi product Legendrian immersion of a point and the totally geodesic Lege…

Whitening, or sphering, is a common preprocessing step in statistical analysis to transform random variables to orthogonality. However, due to rotational freedom there are infinitely many possible whitening procedures. Consequently, there is a diverse range of sphering methods in use, for example based on principal com…

Veronese minimizes normal curvatures to sphere.

The study connects spheres in specific surface curve graphs, proving connectivity and classifying components.

Existence of harmonic maps from higher-dimensional manifolds to spheres proven.

Optimizes sharp curvature inequality on spheres, proving near-minimizers are close to standard metric.

Lower bounds on geodesic lengths for spheres with Willmore energy.

Optimal inequality on sphere for convex bodies.

In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this paper, the Jacobian determinant of a gradient map is shown to be log-concave with respect to a convex combination of the potential functions when the underlying manifold is t…

In this paper, firstly, inspired by Natário's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the sphere $\SS^{n+1}$. We also get the rigidity in the spherical case. Secondly, we use …

NWoS solves high-dimensional Poisson equations using neural networks.

Optimally stabilizes Möbius group maps in spheres across dimensions.

Ricci flow preserves standard sphere's curvature for certain conditions.

We study the min-max optimization problem where each function contributing to the max operation is strongly-convex and smooth with bounded gradient in the search domain. By smoothing the max operator, we show the ability to achieve an arbitrarily small positive optimality gap of $δ$ in $\tilde{O}(1/\sqrtδ)$ computation…

Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.

Three training regimes found for scale-invariant neural networks on the sphere.

Bayesian framework for sphere regression using Gaussian fields.

In a previous paper, we proved a number of optimal rigidity results for Riemannian manifolds of dimension greater than four whose curvature satisfy an integral pinching. In this article, we use the same integral Bochner technique to extend the results in dimension three. Then, by using the classification of closed thre…

New inequality on sphere generalizes circle inequality.

New surfaces near a sphere violate Minkowski inequality.

Improved persistence spheres map measures to functions, stable under partial transport.

In this note we provide natural optimal geometric conditions for a Riemannian manifold suitably covered by two open metric balls to be homeomorphic to a sphere. This can be viewed as a geometric analogue of Brown's theorem in topology stating that a closed manifold covered by two topological balls is a sphere.

A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with $c\ge0$. Making use of the Hamilton-Brendle-Schoen convergence result for Ricci…

The present paper extends the classical second-order variational problem of Herglotz type to the more general context of the Euclidean sphere S^n following variational and optimal control approaches. The relation between the Hamiltonian equations and the generalized Euler-Lagrange equations is established. This problem…

For decades, the sphere eversion has been a classic subject for mathematical visualization. The 1998 video "The Optiverse" shows geometrically optimal eversions created by minimizing elastic bending energy. We contrast these minimax eversions with earlier ones, including those by Morin, Phillips, Max, and Thurston. The…

Sphere theorems proved for manifolds with specific curvature conditions.

This article addresses regularity of optimal transport maps for cost="squared distance" on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved [KM2]. Under boundedness and non-vanishing assumptions on th…

Proves existence of maximizers for eigenvalue optimization on manifolds.

This paper embeds surfaces in 3D spheres and balls with minimal area.

In this paper, we study the first eigenvalue of Jacobi operator on an $n$-dimensional non-totally umbilical compact hypersurface with constant mean curvature $H$ in the unit sphere $S^{n+1}(1)$. We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature $H$ and …

Optimizes energy of mappings from complex projective spaces.

The paper optimizes Dirac eigenvalues on surfaces and connects them to harmonic maps into complex projective spaces.

The study characterizes homology 4-manifolds with $g_2\leq 5$ combinatorially.

In recent years, spectral clustering has become a standard method for data analysis used in a broad range of applications. In this paper we propose a new class of algorithms for multiway spectral clustering based on optimization of a certain "contrast function" over the unit sphere. These algorithms, partly inspired by…