New theorem proves uniqueness of Schwarzschild-de Sitter spacetime.
problem Proving the uniqueness of Schwarzschild-de Sitter spacetime.
method Developed new tools like reverse Lojasiewicz inequality and proved smoothness of maximum-set of lapse.
result Established a new uniqueness theorem for three-dimensional Schwarzschild-de Sitter metrics.
Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.
problem Finite-time singularities of harmonic map flow in critical dimensions.
method Proving a weighted Lojasiewicz inequality.
result Continuity of body map and no-neck property for bubble-tree decompositions.
The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, we first give an elementary geometric, coordinate-based proof of t…
We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasie…
Study proves Lojasiewicz inequalities for harmonic maps near simple bubble trees.
problem Analyzing harmonic maps near simple bubble trees.
method Proves Lojasiewicz inequalities for harmonic maps close to simple bubble trees.
result Obtains new results on the convergence of harmonic map flow and energy spectrum.
We apply our abstract gradient inequalities developed by the authors in arXiv:1510.03817 to prove Lojasiewicz--Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Our Lojasiewicz--Simon gradient …
In this sequel to arXiv:1510.03817, we apply our abstract Lojasiewicz-Simon gradient inequality to prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Lojasiewicz-Simon gradient …
Uniqueness of nondegenerate blowups for planar networks shown.
problem Uniqueness of nondegenerate blowups for the motion by curvature of planar networks.
method Proof based on Lojasiewicz-Simon gradient inequality applied to stability properties of critical points of the length functional.
result Uniqueness of nondegenerate compact blowups for the motion by curvature of planar networks.
This paper explores geometric and analytic aspects of Lojasiewicz inequalities on vector bundles.
problem Understanding growth and stability conditions for real-analytic functions over vector bundles.
method Outline theory of functionals and variational problems over vector bundles, explore applications to real-analytic functionals.
result Describes the energy functional on $S^{n-1$ as a functional over a vector bundle.
The paper proves a Lojasiewicz inequality for maps from the 2-sphere to itself.
problem Analyzing maps from the 2-sphere to itself using Lojasiewicz inequalities.
method Using Lojasiewicz-Simon inequalities and Topping's repulsion estimates, along with a bubble-tree induction argument.
result Polynomial convergence of weak solutions of harmonic map flow on compact domains.
It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Loja…
The elastic flow of curves converges smoothly to a critical point.
problem Smooth convergence of elastic flow of curves.
method Application of Lojasiewicz-Simon inequality.
result Smooth convergence to a critical point.
For any compact Lie group G and closed, smooth Riemannian manifold (X,g) of dimension d≥2, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal G-bundle over X supporting a connection with Lp-small curvature, when p>d/2, to the case of a connection with …
Study on stability of ALE Ricci-flat metrics using a modified Perelman's λ-functional.
problem Stability and instability of ALE Ricci-flat metrics.
method Use of a modified Perelman's λ-functional and Lojasiewicz inequality.
result Demonstrates dynamical instability of ALE Ricci-flat metrics.
The Willmore flow preserves surface volume, leading to convergence to a sphere.
problem Long-term behavior of volume-preserving Willmore flow on surfaces.
method Volume-preserving Willmore flow, blow-up analysis, constrained Lojasiewicz-Simon inequality.
result Smooth solutions exist for spherical surfaces with Willmore energy below 8π and converge to a sphere.
In real algebraic geometry, Lojasiewicz's theorem asserts that any integral curve of the gradient flow of an analytic function that has an accumulation point has a unique limit. Lojasiewicz proved this result in the early 1960s as a consequence of his gradient inequality. Many problems in calculus of variations are que…
New functional proves mass positivity for ALE metrics.
problem Proving mass positivity for ALE metrics with Ricci-flat deformations.
method Introduced a new functional λALE and proved its monotonicity and Lojasiewicz-Simon inequality. result Established that small perturbations of Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass.
Gradient descent with biased rounding errors converges faster under certain conditions.
problem Stagnation or negative impact of rounding errors in neural network training with low precision.
method Analysis of gradient descent with stochastic fixed-point rounding errors under the Polyak-Lojasiewicz inequality.
result Biased rounding errors can improve convergence rates, especially when the Polyak-Lojasiewicz inequality holds.
Paper proves strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
problem Proving strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
method Established Lojasiewicz inequality for pointed W-entropy under cylindrical geometry assumption. result Strong uniqueness of cylindrical tangent flows at first singular time of Ricci flow proved.
Unique cylindrical tangent cone for Simons' hypersurface found.
problem Uniqueness of cylindrical tangent cones for area-minimizing hypersurfaces.
method Developed a new Lojasiewicz inequality for non-isolated singularities.
result Cylindrical tangent cone for Simons' hypersurface is unique.
The article analyzes the stability of a curve shortening flow for planar networks.
problem Stability analysis of anisotropic curve shortening flow for planar networks.
method Used Lojasiewicz-Simon gradient inequality to derive stability results.
result For initial data close to an energy minimizer, the flow exists globally and converges to a different energy minimum.
Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.
problem Understanding the nature and behavior of singularities in level set flow.
method Analytical approach using Lojasiewicz inequality and curvature blow-up rates.
result The arrival time is C2 near a critical point if and only if it satisfies a Lojasiewicz inequality. Sharp stability results for reverse isoperimetric inequalities in 2D.
problem Reverse isoperimetric inequalities in the plane.
method Stability analysis of λ-convex bodies and convex bodies with smooth boundaries. result Sharp stability results for reverse isoperimetric inequalities, including inradius and Cheeger inequalities.
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) versio…
The paper proves stability and convergence of minimal networks under curvature motion.
problem Stability and convergence of minimal networks under curvature motion.
method Proved Lojasiewicz-Simon gradient inequalities for minimal networks.
result Motion by curvature starting from networks close to minimal ones exists for all times and smoothly converges.
In this note, we prove an L2n-energy gap result for Yang-Mills connections on a principal G-bundle over a compact manifold without using Lojasiewicz-Simon gradient inequality (arXiv:1502.00668).
Proof of reverse isoperimetric inequality for black holes.
problem Reverse isoperimetric inequality for black holes in Einstein gravity.
method Geometric-analytic approach.
result Reversal of the usual isoperimetric inequality is explained by curved backgrounds governed by Einstein's equations.
We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the Γ-calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev…
The paper proves smooth convergence of evolving hypersurfaces to critical points.
problem Analyzing the convergence of evolving hypersurfaces.
method Gradient flow of a functional with a Lojasiewicz-Simon inequality.
result Asymptotic convergence to critical points of the functional.
The paper shows mean curvature flow keeps diameter bounded under certain conditions.
problem Proving the bounded diameter of hypersurfaces under mean curvature flow.
method Use of Lojasiewicz inequalities and solution of mean-convex neighbourhood conjecture.
result The intrinsic diameter stays uniformly bounded as the flow approaches the first singular time.
The paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
problem Proving strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
method Establishing a Lojasiewicz inequality for the pointed W-entropy in Ricci flow under the assumption of geometry near the base point being close to a generalized cylinder. result Proves strong uniqueness of generalized cylindrical tangent flows and shows that the subset of points with rectifiable Sqck(N) is horizontally parabolic. The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.
problem Analyzing the negative gradient flow of the Willmore energy plus volume.
method Proved a quantitative reverse isoperimetric inequality and applied it to the flow.
result Initial surfaces converge to a round point in finite or infinite time.
Established a Hardy inequality on Finsler manifolds.
problem Hardy inequality on Finsler manifolds.
method Used geometric properties of Finsler structures to prove the inequality.
result Depends on reversibility constant and uniformity constant of Finsler structure.
Unified framework for complex, split-complex, and dual numbers.
problem Analytic and geometric scope of real-analytic functions.
method Generalized Cauchy-Riemann structure and unified real algebra family.
result Milnor-Le type fibration theorem for nondegenerate algebras.
Study of non-convex potential functions in deep learning with Poincaré inequality.
problem Understanding convergence of stochastic dynamics in non-convex potential landscapes.
method Introduced log-Polyak-Lojasiewicz (log-PL) measures and analyzed their convergence properties.
result Langevin dynamics converges at a rate of O~(1/ε) for sufficiently small ε. New optimal isosystolic inequality found for Finsler reversible 2-tori.
problem Optimal isosystolic inequalities on Finsler tori.
method Survey and new inequality derived from prior work.
result Busemann-Hausdorff area of a Finsler reversible 2-torus with unit systole is at least π/4.
New findings show functional inequalities fail on Finsler manifolds with positive S-curvature.
problem Failure of functional inequalities on Finsler manifolds with positive S-curvature.
method Analysis of Finsler metric measure manifolds with reversibility, flag curvature, and S-curvature.
result Functional inequalities fail on Finsler manifolds with positive S-curvature.
Sharp Hardy and spectral gap inequalities found on special irreversible Finsler manifolds.
problem Understanding Hardy and spectral gap inequalities on irreversible Finsler manifolds.
method Finslerian extension of the method of Riccati pairs.
result Sharpness of Hardy and spectral gap inequalities on specific Finsler manifolds.
We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.
New inequality for eigenfunctions on curved spaces.
problem Eigenfunctions on non-smooth spaces with Ricci curvature.
method Sharp reverse-Hölder inequality for Dirichlet Laplacian eigenfunctions.
result Generalizes classical comparison theorem to curved spaces.
We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in two-dimensional Alexandrov spaces of curvature ⩾c. We also study the equality case.
Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.
problem Establishing reverse Hölder inequalities on Kähler metrics of Fano varieties.
method Using log-concavity and properties of Ricci potentials, the inequality is proven for Fano manifolds with log terminal singularities.
result The inequality holds for Fano varieties with log terminal singularities and the constant depends only on p and the dimension of X.
SAIL-RevKL improves SAIL's convergence by regularizing the objective function.
problem Convergence of self-improving online LLM alignment algorithms.
method Proposed SAIL-RevKL, a regularized objective function to improve optimization landscape.
result Proved SAIL-RevKL satisfies the Polyak-Lojasiewicz (PL) condition with near-linear sample complexity.
Defines signed quasiregular curves and proves growth theorem.
problem Understanding growth of signed quasiregular curves.
method Proves weak reverse Hölder inequality and uses it to prove growth theorem.
result Proves growth theorem for signed quasiregular curves.
AdamL optimizes deep learning models by incorporating loss function information.
problem Adaptive optimizers can suffer from poor generalization due to nonuniform gradient scaling.
method AdamL is a novel adaptive optimizer that considers loss function information for better generalization.
result AdamL achieves faster convergence or lower objective function values compared to other optimizers.
We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t μ- P_t ν\|_1 \leq \frac{W_1(μ,ν)}{\sqrt{t}} \] for all probability measures μ,ν where Pt is the heat semigroup and W1 is the ℓ1-Wasserstein distance. This turns out to be an equivalent formulation of a version of…
The paper proves inequalities for Steklov eigenvalues on finite graphs.
problem Eigenvalues of Laplacians for reversible Markov chains and Steklov eigenvalues.
method Generalized Cheeger inequalities, convergence results, and resolvent convergence.
result Sharp estimate for the first non-trivial Steklov eigenvalue.