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.
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…
New degenerate free boundary minimal annuli found in spherical caps, challenging uniqueness.
problem Non-uniqueness in spherical caps beyond the hemisphere.
method Analyzing a family of embedded free boundary minimal annuli in geodesic balls.
result Degenerate annuli exist, contradicting the Naff-Zhu uniqueness hypothesis.
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.
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…
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.
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.
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 $…
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.
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 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. 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…
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.
Given a compact Lie subgroup G of the isometry group of a compact Riemannian manifold M with a Riemannian connection ∇, it is introduced a G−symmetrization process of a vector field of M and it is proved that the critical points of the energy functional \[ F(X):=\frac{\int_{M}\left\Vert \nabla X\right\Ve…
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.
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 study classifies equivariant biharmonic maps and proves stability results for certain maps.
problem Classifying and analyzing equivariant biharmonic maps and their stability.
method Generalized biharmonic equation for equivariant maps, improved second variation formula for biharmonic maps.
result No stable proper biharmonic maps with constant square norm of tension field exist from a compact Riemannian manifold into a space form of positive sectional curvature.
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 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. 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 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…
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…
We study immersed, connected, umbilic hypersurfaces in the Heisenberg group Hn with n ≥ 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…
Optimizes VAE hyperparameters for efficient training and manifold discovery.
problem Efficiently optimizing hyperparameters in VAEs for complex data.
method Latent Bayesian Optimization (zBO) for hyperparameter trajectory optimization.
result Demonstrated improved performance in finding joint rotationally invariant representations.
Sharp inequalities found for orbifold metrics.
problem Bounding systolic ratios on rotationally symmetric orbifolds.
method Analyzing spindle orbifolds and Besse metrics.
result Upper bounds on systolic ratios are attained at Besse metrics.
New steady solitons found with SO(3) symmetry.
problem Finding steady solitons with specific symmetries.
method Proved existence of a family of solitons with SO(3) symmetry.
result Existence of a one-parameter family of solitons.
New theorems on compactness and finiteness for specific types of self-shrinkers.
problem Characterizing rotationally symmetric self-shrinkers with constraints.
method Compactness and finiteness theorems for self-shrinkers with specific symmetries and constraints.
result Existence of entropy minimizing self-shrinkers diffeomorphic to S1imesSn−1 for each n≥2. We investigate existence and stability of rotationally symmetric critical immersions of variational problems of higher order which were considered by Nitsche.
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…
Variational approximations for curve flows on Riemannian manifolds.
problem Approximating solutions to curvature and elastic flow problems on Riemannian manifolds.
method Variational formulations, finite element approximations, piecewise linear elements, stability analysis.
result Derived schemes can compute rotationally symmetric self-shrinkers and geodesics.
Estimates for p-capacities on symmetric manifolds.
problem Estimating relative p-capacities on symmetric manifolds. method Rotationally symmetric manifolds and novel volumetric estimates.
result Sharp weak (p,q)-embeddings and precise lower bounds of principal p-frequencies.