arXiv research

Rotationally invariant Ricci flows are constructed and shown to converge to spacetimes.

We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains $Ω\subset \mathbb{R}^n$. For a rotationally invariant Cheeger set $C$, the free boundary $\partial C \cap Ω$ consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show…

New AMP algorithms for rotationally invariant models with reduced complexity.

Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.

We give a full classification of complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres: they are either Clifford tori, which are flat, or spheres of Gauss curvature $K \geq K_0$ for a positive constant $K_0$, which we determine explicitly and depends on the geometry of the ambient Ber…

New Ricci flow solutions found with rotational symmetry and cone-like singularities.

In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids $\cC_a$ (minimal surfaces invariant by rotations) in the Heisenberg group with a left-invariant metric. We show that these catenoids have Morse index at least 3 and we bound the index from above in terms of the parameter $…

New AMP algorithms improve multi-layer signal reconstruction.

In this paper we study the gradient Ricci shrinking soliton equation on rotationally symmetric manifolds of dimension three and higher and prove that the only complete examples of such metrics on $S^n$, $\R{n}$ and $\R{}\times S^{n-1}$ are, respectively, the round, flat, and standard cylindrical metrics.

New algorithm for signal estimation in noisy matrix models.

New algorithms improve rank one signal estimation from noisy data.

The paper finds new constant $p$-mean curvature surfaces in the Heisenberg group.

Study stabilizes translating solitons in hyperbolic space for MCF.

Study solves overdetermined problems for rotationally invariant Poisson equations in model manifolds.

Rotationally equivariant convolutions improve molecular property prediction.

The flow of a torus by inverse mean curvature keeps total curvature bounded until singularity.

Paper shows rigidity of rotationally symmetric minimal hypersurfaces in 5D spaces.

The paper classifies rotationally symmetric extremal Kähler metrics on complex manifolds.

In the present paper, we find a system of non-linear ODEs that gives rotationally invariant solutions to the Kapustin-Witten equations in 4-dimensional Euclidean space. We explicitly solve these ODEs in some special cases and find decaying rational solutions, which provide solutions to the Kapustin-Witten equations. Th…

Study classifies and characterizes translators in hyperbolic static universe.

The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.

In this paper we study sets in the $n$-dimensional Heisenberg group $\hhn$ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in $\hhn$. We define a notion of mean curvature for hypersurfaces and we show that the boundary of a…

In this paper we study solitons invariant with respect to the flow generated by a complete Killing vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product $(\mathbb{R} \times P, {\rm d}t^2+g_0)$ and the Killing field is $X=\partial_t$. Similarly to what h…

Combines PCA and AMP for better signal estimation in noisy data.

Researchers classify and describe $K^α$-translators in Euclidean space.

Rotationally symmetric solutions persist after mean curvature flow starts from a double cone.

The paper constructs many ancient solutions to the Yamabe flow on spheres.

Our aim is to study invariant hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, whose mean curvature is given as a linear function in the unit sphere $\mathbb{S}^n$ depending on its Gauss map. These hypersurfaces are closely related with the theory of manifolds with density, since their weighted mean cu…

Study elliptic Weingarten surfaces in warped product space with specific curvature conditions.

The paper classifies hypersurfaces in Heisenberg groups with rotational symmetry.

Rotation invariant algorithms fail on sparse problems even with noise.

Theorems and techniques to form different types of transformationally invariant processing and to produce the same output quantitatively based on either transformationally invariant operators or symmetric operations have recently been introduced by the authors. In this study, we further propose to compose a geared rota…

In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to $O(n)$-invariant co…

In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/λ_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower estimates for the series $\sum λ_{i}^{-2}(Ω)$ where $Ω$ is an extrinsic ball of a proper m…

We study immersed, connected, umbilic hypersurfaces in the Heisenberg group $H_{n}$ with $n$ $\geq$ $2.$ We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigm…

Sharp inequalities found for orbifold metrics.

Optimizes VAE hyperparameters for efficient training and manifold discovery.

We investigate existence and stability of rotationally symmetric critical immersions of variational problems of higher order which were considered by Nitsche.

New steady solitons found with SO(3) symmetry.

We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group Sl(2, R). In particular, all constant mean curvature spheres in those spaces are described explicitly, proving that they are not always embedded. Besides new examples of Dela…

Estimates for $p$-capacities on symmetric manifolds.

In this paper, we show that the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.

Paper proves March's criterion for transience on symmetric manifolds.

We study stable constant mean curvature (CMC) hypersurfaces $Σ$ in slabs in a product space $M\times\r,$ where $M$ is an orientable Riemannian manifold. We obtain a characterization of stable cylinders and prove that if $Σ$ is not a cylinder then it is locally a vertical graph. Moreover, in case $M$ is $\h^n,\r^n$ or $…

Formula for heat coefficients of curved conic singularities derived from Riemannian metrics.

New method for cross-validation in high-dimensional data with dependent or heavy-tailed covariates.

New comparison theorems for rotationally symmetric self-shrinkers help in proving the uniqueness of the Angenent torus.

We propose Cormorant, a rotationally covariant neural network architecture for learning the behavior and properties of complex many-body physical systems. We apply these networks to molecular systems with two goals: learning atomic potential energy surfaces for use in Molecular Dynamics simulations, and learning ground…