arXiv research

Minimal surfaces with negative curvature found in large spheres.

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

The large-N limit of Segal-Bargmann transform on spheres is studied.

The paper extends kernel ridge regression to product kernels and reveals new convergence behaviors.

Study shows how 3+1D cosmologies can evolve to de Sitter space under certain conditions.

We define a large class of integrable nonlinear PDE's, \emph{$k$-symmetric AKS systems}, whose solutions evolve on finite dimensional subalgebras of loop algebras, and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the soluti…

Free boundary minimal submanifolds with boundaries on concentric spheres

We construct a large class of pathological $n$-dimensional topological spheres in ${\mathbb R}^{n+1}$ by showing that for any Cantor set $C\subset {\mathbb R}^{n+1}$ there is a topological embedding $f:{\mathbb S}^n\to{\mathbb R}^{n+1}$ of the Sobolev class $W^{1,n}$ whose image contains the Cantor set $C$.

We compute lower bounds for the Morse index and nullity of constant mean curvature tori of revolution in the three-dimensional unit sphere. In particular, all such tori have index at least five, with index growing at least linearly with respect to the number of the surfaces' bulges, and the index of such tori can be ar…

It is known that any target function is realized in a sufficiently small neighborhood of any randomly connected deep network, provided the width (the number of neurons in a layer) is sufficiently large. There are sophisticated theories and discussions concerning this striking fact, but rigorous theories are very compli…

We present a method for training multi-label, massively multi-class image classification models, that is faster and more accurate than supervision via a sigmoid cross-entropy loss (logistic regression). Our method consists in embedding high-dimensional sparse labels onto a lower-dimensional dense sphere of unit-normed …

Characterizes values at infinity for real polynomial maps with 2D fibers.

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

In this paper we investigate the flow of surfaces by a class of symmetric functions of the principal curvatures with a mixed volume constraint. We consider compact surfaces without boundary that can be written as a graph over a sphere. The linearisation of the resulting fully nonlinear PDE is used to prove a short time…

Estimate sphere area in Sol group up to a factor of 10.

Authors construct hypertori with constant negative mean curvature in a sphere.

Characterizes kernel interpolation in large dimensions, revealing optimal and sub-optimal regions.

New MCMC methods map high-dimensional problems to spheres for better mixing.

The article provides formulas for the number of terms in connected sums of sphere products associated with dual-neighborly polytopes.

Optimizes Dirac eigenvalue bound using curvature and quasi-spherical metrics.

Calegari's 4-spheres from fibered knots are proven standard.

New research finds 145 infinite families of CS spheres are standard.

Arboricity of manifolds is explored, with specific results for 2D surfaces.

Study finds infinite families of Sasaki-Einstein metrics on spheres.

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

Study shows bounded cohomology vanishes for higher dimensional sphere diffeomorphisms.

3D BF theory on certain 3-manifolds evaluated via residues and large k limits.

Deep networks can perfectly classify two low-dimensional manifolds on a sphere with large depth and width.

In this paper we construct families of homology spheres which bound 4-manifolds with intersection forms isomorphic to $-E_8$. We show that these families have arbitrary large correction terms. This result says that among homology spheres, the difference of the maximal rank of minimal sub-lattice of definite filling and…

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

New Einstein metrics found on a 10-dimensional sphere.

Two-parameter models can learn high-dimensional targets via gradient flow.

Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.

Thurston's sphere packing on a 3-dimensional manifold is a generalization of Thusrton's circle packing on a surface, the rigidity of which has been open for many years. In this paper, we prove that Thurston's Euclidean sphere packing is locally determined by combinatorial scalar curvature up to scaling, which generaliz…

We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and a round 4-sphere.

Study finite group actions on exotic aspherical space forms.

We prove a Chern-Lashof type formula computing the expected number of critical points of smooth function on a smooth manifold $M$ randomly chosen from a finite dimensional subspace $V\subset C^\infty(M)$ equipped with a Gaussian probability measure. We then use this formula this formula to find the asymptotics of the e…

Proves a special case of the Gaussian kinematic formula using large sphere limits.

Four minimal spheres found in sphere with special metric.

The spaces of harmonic maps of the projective plane to the four-dimensional sphere are investigated in this paper by means of twistor lifts. It is shown that such spaces are empty in case of even harmonic degree. In case of harmonic degree less than 6 it was shown that such spaces are path-connected and an explicit par…

Study classifies equidistant decompositions in 2D spaces.

The four-dimensional sphere is uniquely rigid in terms of scalar curvature.

New minimal surfaces found in 3D sphere with low genus.

We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson equation. We obtain a classification theorem for bi-umbilical hypersurfaces of t…

In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary…

For odd-dimensional spheres, there's always a second short geodesic.

We show that two of the Bryant-Salamon G_2-manifolds have a simple topology ; homeomorphic to the complement of some submanifolds of the 7-dimensional sphere. In this connection, we show there exists a complete Ricci-flat (non-flat) metric on the complement of an m-dimensional sphere in an n-dimensional sphere for some…

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in th…