Investigates Cheeger sets in rotationally invariant domains and their free boundaries.
problem Properties of Cheeger sets in rotationally invariant domains.
method Analyzes properties of Cheeger sets and their free boundaries, using Delaunay surfaces and constant mean curvature.
result For convex domains, free boundaries consist of spheres and nodoids; for nonconvex domains, unduloids or cylinders can also appear.
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 $…
Study solves overdetermined problems for rotationally invariant Poisson equations in model manifolds.
problem Solving overdetermined problems for rotationally invariant Poisson equations in model manifolds.
method Analyzes specific cases of overdetermined problems and uses geometric properties of model manifolds to deduce radial solutions.
result Conditions on f, φ and κ imply that the solution u is radial and the domain Ω is a geodesic ball centered at O. Rotationally invariant Ricci flows are constructed and shown to converge to spacetimes.
problem Constructing and understanding Ricci flows through surgery on rotationally invariant manifolds.
method Rotationally invariant Ricci flow through surgery, convergence to spacetimes, blowup rate analysis.
result Rotationally invariant Ricci flows converge to spacetimes with controlled curvature blowup.
In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient ∑1/λirad=∫V(s)/S(s)ds. We also obtain upper and lower estimates for the series ∑λi−2(Ω) where Ω is an extrinsic ball of a proper m…
Study classifies and characterizes translators in hyperbolic static universe.
problem Classifying and characterizing translators in hyperbolic static universe.
method Classified and characterized translators foliated by horospheres and rotationally invariant ones, both space-like and time-like.
result Obtained a characterization of the bowl and certain translators foliated by horospheres.
The study proves geometric inequalities for static convex domains in static rotationally symmetric spaces.
problem Proving geometric inequalities for static convex domains in static rotationally symmetric spaces.
method Locally constrained curvature flow in a static rotationally symmetric space Nn+1, proving graphical solutions and static convexity preservation. result Proves weighted geometric inequalities for static convex domains close to a slice of Nn+1. New AMP algorithms for rotationally invariant models with reduced complexity.
problem Signal estimation in generalized linear models with arbitrary spectral design matrices.
method Rotationally invariant approximate message passing (AMP) algorithms.
result Performance close to Vector AMP with significantly lower complexity.
Classification of constant curvature surfaces in Berger spheres.
problem Identifying complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres.
method Complete classification through detailed analysis of Clifford tori and spheres.
result Rotationally invariant spheres with constant Gauss curvature are the only topological spheres in Berger spheres for K>KP. New findings on magnetic geodesic flows and periodic motions.
problem Characterizing superintegrable systems in magnetic geodesic flows.
method Analyzing rotationally symmetric magnetic geodesic flows.
result All sufficiently slow motions in a central magnetic field are periodic under specific curvature and homogeneity conditions.
Let Ω be a star-shaped bounded domain in (Sn,ds2) with smooth boundary. In this article, we give a sharp lower bound for the first non-zero eigenvalue of the Steklov eigenvalue problem in Ω. This result is the generalization of a result given by Kuttler and Sigillito for a star-shaped bounded doma…
Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.
problem Bounding Laplacian eigenvalues on manifolds with non-negative scalar curvature.
method Investigation of invariant spectrum on compact Riemannian manifolds with large isometry groups.
result Upper bounds for eigenvalues of the invariant spectrum assuming non-negative scalar curvature.
New Ricci flow solutions found with rotational symmetry and cone-like singularities.
problem Finding Ricci flow solutions with specific symmetry and singularity properties.
method Rotationally symmetric Ricci flow with scaling-invariant curvature bounds, using approximation method.
result Complete Ricci flow solution with cone-like singularity at the origin.
Study invariant hypersurfaces with linear mean curvature in Euclidean space.
problem Understanding hypersurfaces with linear mean curvature.
method Explicit parametrizations and classification of rotationally invariant hypersurfaces.
result Obtained explicit parametrizations of constant curvature hypersurfaces.
New AMP algorithms improve multi-layer signal reconstruction.
problem Reconstructing signals and hidden variables from multi-layer networks with rotationally invariant weights.
method Developed multi-layer rotationally invariant generalized AMP (ML-RI-GAMP) algorithms and state evolution recursion.
result ML-RI-GAMP outperforms existing methods in terms of lower complexity and similar performance.
Formula for Heisenberg group surface areas derived.
problem Deriving a formula for surface areas in Heisenberg groups.
method Analogy of Cauchy's surface area formula in Heisenberg groups.
result Formula for p-area of compact hypersurfaces in Heisenberg groups.
We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show …
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 Sn, Rn and R×Sn−1 are, respectively, the round, flat, and standard cylindrical metrics.
New algorithm for signal estimation in noisy matrix models.
problem Signal estimation in rectangular spiked matrix models with rotationally invariant noise.
method Orthogonal Approximate Message Passing (OAMP) algorithm for signal estimation.
result Optimal OAMP algorithm minimizes mean-squared error and achieves Bayes-optimal performance.
Cormorant learns molecular properties via rotationally covariant neural networks.
problem Learning molecular potential energy surfaces and properties.
method Rotationally covariant neural network architecture with tensor products and Clebsch-Gordan decomposition.
result Significantly outperforms competing algorithms in learning molecular Potential Energy Surfaces.
New algorithms improve rank one signal estimation from noisy data.
problem Estimating a rank one signal matrix from corrupted data with rotationally invariant noise.
method Developed approximate message-passing algorithms exploiting eigenvalues and iterates denoisers.
result Achieves optimal asymptotic estimation error among iterative algorithms.
The paper finds new constant p-mean curvature surfaces in the Heisenberg group.
problem Discovering new examples of constant p-mean curvature surfaces. method Utilizing the theory and approach for constructing such surfaces.
result Complete description of rotationally invariant surfaces of constant p-mean curvature. Study stabilizes translating solitons in hyperbolic space for MCF.
problem Stability of translating solitons in hyperbolic space.
method Developed theory, constructed rotationally invariant translators, used avoidance principle and maximum principle.
result Horospheres are dynamically stable as radial graphical solutions to MCF.
Rotationally equivariant convolutions improve molecular property prediction.
problem Predicting molecular properties using graph neural networks.
method Ablation study with rotationally equivariant and invariant convolutions on QM9 data set.
result Rotationally equivariant layers decrease test error by an average of 23%.
The flow of a torus by inverse mean curvature keeps total curvature bounded until singularity.
problem Understanding the behavior of a torus under inverse mean curvature flow until singularity.
method Analyzing the evolution of a rotationally symmetric embedded torus in R3 by inverse mean curvature flow. result The total curvature remains bounded until the singular time Tmax. We provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely…
Let u denote a solution to a rotationally invariant Hessian equation F(D2u)=0 on a bounded simply connected domain Ω⊂R2, with constant Dirichlet and Neumann data on ∂Ω. In this paper we prove that if u is real analytic and not identically zero, then u is radial and Ω is a disk. The fully …
Paper shows rigidity of rotationally symmetric minimal hypersurfaces in 5D spaces.
problem Understanding rigidity of rotationally symmetric minimal hypersurfaces in 5D spaces.
method Pointwise hypersurface invariant analysis for minimal hypersurfaces in spaces of constant curvature.
result Rotationally symmetric minimal hypersurfaces in 5D spaces are rigid.
The paper classifies rotationally symmetric extremal Kähler metrics on complex manifolds.
problem Classifying extremal Kähler metrics on complex manifolds.
method Analyzing polynomial zeros in Calabi's extremal equation.
result No U(n) invariant complete extremal Kähler metrics on Cn with positive bisectional curvature. 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…
The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two m-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of rotationally symmetric, proper biharmonic conformal diffeomorphisms in the special case th…
The study examines special domains in S^2 supporting specific solutions to a PDE.
problem Identifying special domains in S^2 supporting positive solutions to a PDE.
method Extends moving plane method and Alexandrov reflection method to prove symmetry.
result Domains must be rotationally symmetric under specific conditions.
The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.
problem Maximizing the first normalized Laplace-Beltrami eigenvalue on tori.
method Constructing equivariant harmonic maps to spheres and analyzing their properties.
result Rotationally symmetric critical metrics for the first eigenvalue are found and characterized.
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 (R×P,dt2+g0) and the Killing field is X=∂t. Similarly to what h…
Combines PCA and AMP for better signal estimation in noisy data.
problem Estimating a rank-1 signal in rotationally invariant noise.
method Combines PCA and AMP, with PCA initialization at the start of AMP.
result Rigorous asymptotic characterization of the new estimator's performance.
Researchers classify and describe Kα-translators in Euclidean space.
problem Classifying and describing Kα-translators in Euclidean space. method Rotationally symmetric and helicoidal motions.
result For each α, there is a Kα-translator intersecting orthogonally the rotation axis. Rotationally symmetric solutions persist after mean curvature flow starts from a double cone.
problem Understanding the symmetry of solutions to mean curvature flow.
method Analyzing solutions coming out of a double cone.
result Rotationally symmetric solutions persist.
The paper constructs many ancient solutions to the Yamabe flow on spheres.
problem Ancient solutions to the Yamabe flow on spheres.
method Non-radial inner--outer gluing scheme, conformal invariance, weighted Hölder estimates.
result Uncountably many non-rotationally symmetric ancient solutions.
Study elliptic Weingarten surfaces in warped product space with specific curvature conditions.
problem Characterize elliptic Weingarten surfaces in warped product spaces with minimal type curvature conditions.
method Analyze surfaces with mean curvature and extrinsic curvature satisfying a specific relationship under radial symmetry of the warping function.
result Existence and uniqueness of rotationally-invariant elliptic Weingarten surfaces of minimal type in RimeshR. The paper proves existence and instability of weak r-harmonic maps.
problem Existence and stability of weak r-harmonic maps. method Construction of critical points and analysis of stability.
result Existence and instability of weak r-harmonic maps restricted to specific dimensions. Paper introduces S3W distance for spherical probability distributions.
problem Comparing spherical probability distributions efficiently and accurately.
method S3W distance using stereographic projection and generalized Radon transform.
result Extensive theoretical analysis and evaluation of S3W performance.
The paper classifies hypersurfaces in Heisenberg groups with rotational symmetry.
problem Classifying hypersurfaces in Heisenberg groups with rotational symmetry.
method Fundamental theorems and earlier results in [3] and [4] were used to classify umbilic hypersurfaces and generate curves for hypersurfaces with constant p-mean curvature. result Complete classification of umbilic hypersurfaces and generating curves in Heisenberg groups Hn. Paper solves Serrin problem for ring-shaped domains, showing velocity has finitely many maxima.
problem Characterizing rotationally symmetric solutions to a specific PDE on a ring-shaped domain.
method Introduced new arguments in the spirit of comparison geometry to overcome the lack of monotonicity.
result Simplest conditions are not sufficient; rotational symmetry requires finitely many maxima.
We prove the existence of a countable family of Delaunay type domains Ω_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 in M^n) for which the first eigenfunction of the Laplace-Beltrami operator with zero …
Rotation invariant algorithms fail on sparse problems even with noise.
problem Rotation invariant algorithms' suboptimality in sparse linear problems with noise.
method Lower bounds and trajectory analysis of optimization algorithms.
result Rotation invariant algorithms are suboptimal even with noise and many examples.
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…