A new optimization method handles Euclidean bounds efficiently.
problem Optimization problems with Euclidean bounds.
method Riemannian limited-memory BFGS method combining quasi-Newton and Riemannian adaptations.
result Outperforms existing methods by several orders of magnitude.
Compactness theorem for manifolds with scalar curvature and entropy bounds.
problem Understanding the structure of manifolds with specific curvature and entropy bounds.
method Using volume upper bounds to prove Gromov-Hausdorff closeness to Euclidean balls.
result Unit balls in such manifolds are bi-Hölder and bi-W1,p homeomorphic to Euclidean balls. This paper tightens the generalization error bound for graph embedding in non-Euclidean spaces.
problem High generalization error in non-Euclidean graph embedding, preventing practical applications.
method Novel upper bound of graph embedding's generalization error using local Rademacher complexity.
result The new bound is tighter and faster, allowing better performance in non-Euclidean spaces.
Survey on complex submanifolds in Euclidean spaces.
problem Existence of complete bounded complex submanifolds.
method Historical overview and recent progress.
result Discussion of open questions in the field.
In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or a Hadamard manifold, and of a biharmonic Steklov problem on bounded domains of a…
Upper bound on geodesic ball volume in Riemannian manifolds.
problem Bounding geodesic ball volume in Riemannian manifolds.
method Techniques to provide an upper bound on geodesic ball volume.
result Upper bound on geodesic ball volume in Euclidean space.
We prove generalized lower Ricci bounds for Euclidean and spherical cones over compact Riemannian manifolds. These cones are regarded as complete metric measure spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-di…
Sharp upper bound found for Steklov spectrum on revolution submanifolds.
problem Finding bounds for Steklov spectrum on specific submanifolds.
method Analyzing submanifolds of revolution in Euclidean space.
result Sharp upper bound established for Steklov spectrum.
In this paper, we use a weighted isoperimetric inequality to give a lower bound on the first Dirichlet eigenvalue of the Laplacian on a bounded domain inside a Euclidean cone. Our bound is sharp, in that only sectors realize it. This result generalizes a lower bound of Payne and Weinberger in two dimensions.
A lower bound for the number of 3-periodical billiard trajectories in a manifold embedded in Euclidean space is obtained.
Optimal bounds found for torus curvatures in high dimensions.
problem Finding optimal bounds on normal curvatures of tori.
method Analyzing immersed n-torus in a Euclidean ball of large dimension.
result Optimal bounds on normal curvatures of tori established.
New upper bound for Neumann Laplacian eigenvalues on convex domains.
problem Bounding Neumann eigenvalues on convex domains.
method Deriving a new upper bound for eigenvalues.
result Universal inequalities for Neumann eigenvalues derived from the upper bound.
Sharp bounds for anisotropic p-capacity of Euclidean compact sets derived using flow methods.
problem Sharp bounds for anisotropic p-capacity of Euclidean compact sets.
method Inverse anisotropic mean curvature flow (IAMCF) and anisotropic Hawking mass.
result Upper bounds for anisotropic p-capacity derived using flow methods.
Upper bounds for Steklov eigenvalues derived from intersection indices.
problem Finding upper bounds for Steklov eigenvalues of submanifolds in Euclidean space.
method Using intersection indices of submanifolds and their boundaries.
result Explicit upper bounds involving intersection index, volume, and dimensional constants.
Sharp upper bounds found for Steklov eigenvalues of a specific hypersurface.
problem Finding upper bounds for Steklov eigenvalues of a specific type of hypersurface.
method Analytical approach to compute upper bounds and prove stability properties.
result Sharp upper bounds Bn(L) and Bn for Steklov eigenvalues are derived. The paper proves pseudolocality theorems for Ricci flows on incomplete manifolds.
problem Pseudolocality of Ricci flows on incomplete manifolds.
method Proves pseudolocality theorems for Ricci flows under specific curvature and isoperimetric conditions.
result Constructs solutions of Ricci flow in balls with pseudolocality property.
We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.
New bounds on Laplace operator eigenvalues for submanifolds in Euclidean spaces.
problem Finding upper bounds for the first eigenvalue of the Laplace operator on compact submanifolds.
method Using a new technique, bounds depend on length of mean curvature vector, dimension, volume, and vector in Euclidean space.
result Improved and new upper bounds computed for non-minimally embedded submanifolds.
The paper provides bounds for embedding manifolds into Euclidean spaces with group actions.
problem Finding explicit bounds for embedding manifolds into Euclidean spaces with group actions.
method The paper provides upper and lower bounds for the dimension of the Euclidean space required for equivariant embeddings of manifolds into Euclidean spaces for finite group actions.
result Explicit bounds for the dimension of the Euclidean space required for equivariant embeddings of manifolds into Euclidean spaces for finite group actions.
If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily altern…
Study improves bounds on p-covectors and proves stable systolic inequalities.
problem Improving bounds on p-covectors and proving stable systolic inequalities.
method Analyzing Euclidean norms and fundamental cohomology classes.
result Improved upper bounds and stable systolic inequalities proved.
We prove generalized lower Ricci bounds for Euclidean and spherical cones over complete Riemannian manifolds. These cones are regarded as complete metric measure spaces. In general, they will be neither manifolds nor Alexandrov spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricc…
Sharp lower bound found for area of vector fields on spherical annuli.
problem Finding the minimum area of unit vector fields on spherical annuli.
method Established a sharp lower bound through mathematical analysis.
result Sharp lower bound for the area of unit vector fields on spherical annuli.
The paper proves bounded cohomology properties of Euclidean space groups.
problem Understanding bounded cohomology of transformation groups of Euclidean spaces and discs.
method Analyzes groups of homeomorphisms and diffeomorphisms of Euclidean spaces and discs, proving boundedly acyclic properties.
result Groups of orientation-preserving homeomorphisms and diffeomorphisms of Rn are boundedly acyclic, with implications for characteristic classes of bundles. In this short note we prove that the degree of the Gauss map ν of a closed 3-dimensional hypersurface of the Euclidean space is a lower bound for the total bending functional B, introduced by G. Wiegmink. Consequently, the energy functional E introduced by C. M. Wood admits a topological lower bound.
Paper revisits DP-SCO in Euclidean and ℓpd spaces, focusing on constrained and bounded sets.
problem Differentially private stochastic convex optimization in constrained and bounded sets in Euclidean and ℓpd spaces. method Proposes methods achieving excess population risks dependent on Gaussian width of the constraint set, and novel algorithms for unconstrained and heavy-tailed data.
result Theoretical results for DP-SCO in ℓpd spaces, including optimal bounds for strongly convex functions. Estimates area and spectrum of stable minimal surfaces in Euclidean and hyperbolic spaces.
problem Estimating the growth of area and spectrum of stable minimal surfaces.
method Elementary argument and stability inequality for Euclidean space; explicit area growth estimate for hyperbolic space; scalar curvature lower bound for spectrum.
result Minimal surfaces in Euclidean space grow like the Euclidean plane, and in hyperbolic space, explicit area growth estimates are derived.
Sharp upper bounds derived for capacities in hyperbolic and Euclidean spaces.
problem Finding upper limits for the capacity of compact sets in hyperbolic and Euclidean spaces.
method Inverse mean curvature flow, unit-speed normal flow, weak inverse mean curvature flow, inverse anisotropic mean curvature flow.
result Various sharp upper bounds for the p-capacity of compact sets in hyperbolic and Euclidean spaces are derived. Constructs flows on manifolds with small curvature, proving Euclidean topology.
problem Geometric structure of manifolds with unbounded curvature.
method Distance like functions with integral hessian bound, Ricci flows.
result Manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors n-regular and small curvature concentration are topologically Euclidean. Characterizes kernel of linearization for minimal surfaces problem
problem Characterizing kernel of linearization for minimal surfaces problem
method Show kernel consists of potential fields and TT fields
result In whole-space Euclidean decomposition, kernel consists of potential fields and TT fields
The paper calculates the index and nullity of Fraser-Sargent surfaces and provides bounds.
problem Computing the index and nullity of Fraser-Sargent surfaces.
method Analytical and numerical methods to compute index and nullity; provides bounds.
result Lower and upper bounds on the index of Fraser-Sargent surfaces inside the ball.
Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actu…
We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with conne…
We show that non-collapsed Gromov-Hausdorff limits of polarized Kahler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson-Sun, in…
The study characterizes compact submanifolds with pinched Ricci curvature in Euclidean and spherical space forms.
problem Characterizing compact submanifolds with specific Ricci curvature bounds.
method Proving rigidity results for submanifolds with Ricci curvature bounded below by a function of mean curvature.
result Submanifolds are either isometric to the Einstein Clifford torus or have vanishing homology groups up to a certain degree.
We compute the whole spectrum of the Dirichlet-to-Neumann operator acting on differential p-forms on the unit Euclidean ball. Then, we prove a new upper bound for its first eigenvalue on a domain Ω in Euclidean space in terms of the isoperimetric ratio ${\rm Vol}(\bdΩ)/{\rm Vol}(Ω)$.
The paper examines torsions in Minkowskian product of Finsler metrics.
problem Investigating Cartan torsion and mean Cartan torsion in Minkowskian product of Finsler metrics.
method Deriving explicit formulas for Cartan torsion and mean Cartan torsion, analyzing their geometric behavior, and providing conditions for bounded norm.
result Both Cartan torsion and mean Cartan torsion decompose additively if and only if the Minkowskian product is Euclidean, and a necessary and sufficient condition for the norm of the mean Cartan torsion to remain bounded in the Euclidean case is provided.
In this paper we obtain an analogue of Toponogov theorem in dimension 3 for compact manifolds M3 with nonnegative Ricci curvature and strictly convex boundary ∂M. Here we obtain a sharp upper bound for the length L(∂Σ) of the boundary ∂Σ of a free boundary minimal surface Σ2 in M3 i…
The paper studies minimal graphs with bounded 2-dilation in Euclidean space.
problem Understanding minimal graphs with bounded 2-dilation in Euclidean space.
method Analyzing tangent cones and proving Neumann-Poincaré inequalities.
result Minimal graphs have multiplicity one tangent cones at infinity.
For collapsing sequences of Riemannian manifolds which satisfy a uniform lower Ricci curvature bound it is shown that there is a sequence of scales such that for a set of good base points of large measure the pointed rescaled manifolds subconverge to a product of a Euclidean and a compact space. All Euclidean factors h…
New algorithms optimize convex functions with high-order derivatives.
problem Optimizing convex functions with high-order derivatives under various norms.
method Developed a non-Euclidean inexact accelerated proximal point method using an inexact uniformly convex regularizer.
result Showed nearly optimal algorithms for high dimensions in the black-box oracle model for ℓp-settings and all q≥1. Study on minimal submanifolds with finite curvature in Euclidean space.
problem Finite diffeomorphism types of complete immersed minimal submanifolds with finite total curvature.
method Adapted method from Chodosh, Ketover, and Maximo for hypersurfaces to submanifolds of arbitrary codimension.
result Proved finite diffeomorphism types for complete immersed minimal submanifolds with finite total curvature.
Study bounds derivatives of solutions to a specific equation on domains.
problem Bounding second derivatives of solutions to the σk-Yamabe equation. method Proves local pointwise second derivative estimates for positive W2,p solutions. result Establishes bounds for derivatives of solutions to the σk-Yamabe equation. Study magnetic Schrödinger operators in Euclidean space.
problem Semiclassical spectral analysis of magnetic Schrödinger operators.
method Spectral problems for Bochner-Schrödinger operator on manifolds.
result Survey and describe ideas of proofs for magnetic Schrödinger operators.
We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence…
By a classical result of Jordan, each finite subgroup G of a complex linear group GL_n(C) has an abelian subgroup whose index in G is bounded by a constant depending only on n. We consider the problem if this remains true for finite subgroups G of the diffeomorphism group of a smooth manifold, and show that it is true …
Harmonic maps to Euclidean buildings have rectifiable singular strata.
problem Understanding the structure of singular points for harmonic maps.
method Defining singular strata and proving rectifiability using the rectifiable Reifenberg program.
result Rectifiability of singular strata for harmonic maps into F-connected complexes. Local smoothing of metrics with small curvature, removing Ricci curvature condition.
problem Establishing local smoothing of metrics with curvature concentration.
method Local mollification, removing Ricci curvature condition, Sobolev constants and volume growth.
result Compactness of manifolds with small curvature concentration under Ahlfors regularity and Sobolev constant.