The paper finds solutions to Kapustin-Witten equations with singular imaginary parts.
problem Finding solutions to Kapustin-Witten equations with specific properties.
method Solving a system of non-linear ODEs to find rotationally invariant solutions.
result Explicit solutions found, including decaying rational solutions and singular imaginary parts.
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.
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.
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.
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.
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 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.
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. 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. 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.
We study a second order ordinary differential equation corresponding to rotationally symmetric p-harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive entire solutions. Asymptotic properties of the positive solutions are investiga…
New ancient compact solutions found for Yamabe flow.
problem Finding compact solutions to the Yamabe flow.
method Constructed rotationally symmetric ancient compact solutions.
result Found type I ancient compact solutions converging to self-similar solutions.
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. 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. We study a second order differential equation corresponding to rotationally symmetric F-harmonic maps between certain noncompact manifolds. We show unique continuation and Liouville's type theorems for positive solutions. Asymptotic properties and the existence of bounded positive solutions are investigated.
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…
Study of homothetic solitons in inverse mean curvature flow.
problem Understanding the behavior of solitons in inverse mean curvature flow.
method Analyzing solutions that evolve by homotheties of a given submanifold.
result Classification of rotationally invariant Lagrangian homothetic solitons.
Study on solutions to conformally invariant fourth order equations, classifying their properties.
problem Classify qualitative properties of solutions to conformally invariant fourth order equations.
method Analyze two cases: removable and non-removable singularities, using Pohozaev-type invariant.
result Non-existence of semi-singular solutions, classifying them as multiples of the Emden--Fowler solution.
In this paper we study the classification of ancient convex solutions to the mean curvature flow in Rn+1. An open problem related to the classification of type II singularities is whether a convex translating solution is k-rotationally symmetric for some integer 2≤k≤n, namely whether its level set is a …
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.
Rotationally symmetric solutions for mean curvature flow.
problem Understanding singularities in mean curvature flow.
method Analyzing cylindrical and gradient estimates for translators.
result Rotationally symmetric solutions for a class of flows.
Authors propose GRI-CNN systems for rotationally invariant processing.
problem Creating rotationally invariant CNN systems.
method Designed geared rotationally identical CNN systems (GRI-CNN) with a small step angle.
result GRI-CNN produces quantitatively identical output results under rotation.
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.
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 $…
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…
We study "warped Berger" solutions $\big(\mc S^1\times\mc S^3,G(t)\big)$ of Ricci flow: generalized warped products with the metric induced on each fiber {s}×SU(2) a left-invariant Berger metric. We prove that this structure is preserved by the flow, that these solutions develop finite-time neckpinch …
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.
New formulas for p-capacitary potentials in convex domains.
problem Analyzing geometric properties of p-capacitary potentials.
method Monotonicity formulas derived from p-Laplace equation solutions.
result New characterizations of rotationally symmetric solutions and domains.
Study of constant curvature hypersurfaces in Heisenberg group.
problem Characterizing umbilic hypersurfaces with constant curvature in Heisenberg group.
method Analyzing immersed, connected, umbilic hypersurfaces in Heisenberg group Hn, proving rotation invariance and uniqueness of Pansu spheres. result Pansu spheres are the only rotationally invariant, closed, umbilic hypersurfaces with positive constant sigma-k curvature in Heisenberg group.
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 comparison theorems for rotationally symmetric self-shrinkers help in proving the uniqueness of the Angenent torus.
problem Uniqueness of the Angenent torus in rotationally symmetric self-shrinkers
method Analyzing profile curves and vertical points of rotationally symmetric self-shrinkers
result Proving the existence and monotonicity of horizontal-point trajectories
New methods prove existence of rotating shapes moving in space.
problem Existence of rotating shapes moving in space.
method Different methods to prove existence based on singular ordinary differential equation.
result Existence of rotationally symmetric translating solutions proven without partial differential equations.
Study Ricci flow on Rn+1, focusing on asymptotic behavior and singularities.
problem Analyzing the behavior of Ricci flow on Rn+1, especially near singularities. method Examined the flow starting from rotationally symmetric metrics, considering asymptotic curvature and pinched necks.
result Proved the existence of Type-II and Type-I singularities under specific conditions.
When a Riemannian manifold (M,g) is rotationally symmetric, the critical order of the lower bound of radial curvatures for the absence of eigenvalues of the Laplacian is equal to −r1, where r stands for the distance to the center point. In this paper, we shall perturb the Riemannian metric around a rota…
Constructing solutions to geometric flows with rotational symmetry.
problem Finding solutions to extrinsic geometric flows with specific properties.
method Rotationally symmetric translating solutions constructed for α-homogeneous speeds. result These solutions are necessarily convex and have specific asymptotic behaviors.
New convex ancient solutions found for flows by high powers of curvature.
problem Existence of closed convex ancient solutions to curvature flows.
method Proves existence of closed convex ancient solutions with specific curvature flow speeds.
result Existence of non-homothetic convex ancient solutions for flows by high powers of curvature.
In this work we investigate the following isoperimetric problem: to find the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the area of the part of the boundary contained in the horospheres is not included). We reduce the problem to the study of rotationa…
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.
New proof for rotationally symmetric gradient Ricci solitons in 2-4 dimensions.
problem Existence of rotationally symmetric gradient Ricci solitons in specific dimensions.
method Analytical proof using differential equations.
result Existence and uniqueness of solutions for the given equations.
We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as t→−∞, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.
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. Analyzes solutions to non-elliptic equations on bounded domains.
problem Analyzes solutions to non-elliptic equations on bounded domains.
method Analyzes solutions to non-elliptic equations on bounded domains.
result Analyzes solutions to non-elliptic equations on bounded domains.
The paper studies self-expanding solutions to inverse curvature flows in Euclidean spaces.
problem Investigating self-expanding solutions to inverse curvature flows in Euclidean spaces.
method Using homogeneous symmetric functions of principal curvatures, the paper analyzes self-expanding solutions to a broad class of inverse curvature flows.
result Complete non-compact self-expanders to these flows with asymptotically cylindrical ends must be rotationally symmetric.
The paper studies eigenvalues and Green functions for various domains.
problem Analyzing eigenvalues and Green functions for different domains.
method Using spectral identities, iteration of the Green operator, and explicit formulas.
result Explicit formulas for the first eigenvalue of bounded domains.
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 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.