Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
We prove the two theorems of the title, settling two long standing questions in the local theory of singular minimal hypersurfaces. The sharpness of either result is with respect to its hypothesis on the size of the allowable singular sets. The proofs of both theorems rely heavily on the author's recent regularity and …
Sharp pinching conditions restrict the geometry and topology of submanifolds.
problem Understanding submanifolds under pinching conditions in arbitrary Riemannian manifolds.
method Analyzing submanifolds with pinching conditions involving second fundamental form and mean curvature.
result The pinching condition imposes strong geometric and topological restrictions on submanifolds.
Sharp gradient estimates for positive Ricci curvature manifolds.
problem Understanding geometric properties of manifolds with positive Ricci curvature.
method Proving sharp gradient estimates and monotonicity formulae.
result Sharp gradient estimates and monotonicity formulae for positive Ricci curvature manifolds.
Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.
problem Understanding the p-Laplacian on RCD(K,N) spaces.
method Proving a Talenti-type comparison theorem.
result Sharp, rigid and stable Talenti-type comparison theorem.
Sharp stability of Alexandrov's theorem for C1 domains in the small-excess regime
problem Stability of Alexandrov's theorem for C1 domains in the small-excess regime method Combines a BV version of Fuglede's spectral-gap argument, a star-shaped rearrangement for sets of finite perimeter, quantitative estimates for the part of the boundary contained in the tentacles, and a polyhedral approximation argument for the non-graphical region result Sharp stability estimate in a genuinely non-parametric regime
Let Ω⊂Rd,d≥2, be a bounded open set, and denote by λ_j(Ω),j≥1, the eigenvalues of the Dirichlet Laplacian arranged in nondecreasing order, with multiplicities. The weak form of Pleijel's theorem states that the number of eigenvalues λ_j(Ω), for which there exists an associated eigenf…
Sharp proof of sub-Riemannian length-minimizing curves being at least C2
problem Smoothness of sub-Riemannian length-minimizing curves
method Study of a class of sub-Riemannian structures, proving C2 regularity result Theorem 1.1 in [6] is sharp
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we …
Sharpness of actions on reductive homogeneous spaces proven for various groups.
problem Proving proper and cocompact actions on reductive homogeneous spaces.
method Using quasi-isometric embedding and Anosov representations.
result Characterization and proof of non-compactness for certain homogeneous spaces.
We give sharp sectional curvature estimates for complete immersed cylindrically bounded m-submanifolds φ:M→N×Rℓ, n+ℓ≤2m−1 provided that either φ is proper with the second fundamental form with certain controlled growth or M has scalar curvature with strong quadratic decay. This l…
The study examines vector fields with integer singularities in 3D balls.
problem Characterizing the strong Lp-closure of vector fields with finitely many integer singularities. method Characterization and decomposition of vector fields with finitely many integer singularities.
result Decomposition theorem for elements in LZ1(B), revealing information about mass-minimizing currents. Sharp bounds on hyperbolic metrics in Ptolemaic spaces are derived.
problem Finding sharp bounds on hyperbolic metrics in Ptolemaic spaces.
method Construction of metrics on open subsets of Ptolemaic spaces.
result Sharp parameter bounds for hyperbolic and strongly hyperbolic metrics are derived.
The paper proves curvature-related dimension bounds for manifolds.
problem Proving dimension bounds for manifolds with positive scalar curvature.
method Using asymptotic cones and linear growth harmonic functions.
result Upper bounds on essential and Hausdorff dimensions of manifolds.
Paper proves weak unique continuation for harmonic functions on RCD spaces but finds counterexample for strong uniqueness.
problem Unique continuation of harmonic functions on RCD spaces, especially strong uniqueness.
method Establishes weak unique continuation theorem and provides counterexample for strong uniqueness.
result Found counterexample for strong unique continuation in RCD(K,N) spaces for N≥4 and K∈R.
Study shows distance to boundary is always attained on varifolds with bounded curvature.
problem Understanding varifolds with bounded mean curvature in Riemannian manifolds.
method Proves a barrier principle at infinity using sharp maximum principles.
result Distance to boundary is always attained on varifolds with bounded curvature.
New example solves topological dynamics problem.
problem Embedding compact metric space into cubical shift.
method Borsuk-Ulam theorem, p-adic completions, equivariant Sullivan conjecture.
result Existence of a compact metric space not embeddable into a cubical shift.
Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.
The paper proves gradient and comparison inequalities for RCD spaces.
problem Gradient and comparison inequalities for RCD spaces.
method Elliptic Dirichlet problems and Talenti-type comparison.
result Sharp, rigid, and stable Talenti-type comparison results.
Sharp gradient bound found for compact manifolds.
problem Finding a sharp gradient bound for compact manifolds.
method Using a specific function α=1 in the Li-Yau gradient bound.
result A sharp gradient bound is found for compact manifolds.
In his 1979 paper Trotman proves, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the strong (Whitney) topology implies that the stratification is (a)-regular. Here…
Sharp bounds found for minimal surface solutions.
problem Finding bounds for minimal surface solutions.
method Analyzing minimal surface equation with specific boundary conditions.
result Sharp bounds established for solutions over certain domains.
Sharp spectral theorem splits certain non-compact manifolds.
problem Proving spectral splitting for non-compact manifolds with specific curvature conditions.
method Sharp spectral analysis and geometric splitting theorem.
result Non-compact manifolds split as RimesN under given curvature constraints. Generalized Blaschke rolling theorem for curved spaces.
problem Extending classical theorem to curved spaces.
method Generalization to Riemannian manifolds with bounded curvature.
result Sharp results in arbitrary dimensions, new even in constant curvature spaces.
New method TLC improves transductive learning bounds.
problem Sharp generalization bounds for transductive learning.
method Transductive Local Complexity (TLC) framework.
result Nearly sharp bounds consistent with inductive results.
We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant ver…
The paper explores geometric relationships in manifolds with curvature constraints, proving new inequalities and rigidity results.
problem Understanding geometric features of manifolds with curvature constraints.
method Comparison theorems and spacetime harmonic functions.
result Partial resolution of Gromov's conjecture and new characterizations of geometries.
In this note, we reveal that our solution of Demailly's strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Kollár and Jonsson-Mustată implies the truth of twisted versions of the strong openness conjecture; our optimal L2 extension implies Berndtsson…
We provide an isoperimetric comparison theorem for small volumes in an n-dimensional Riemannian manifold (Mn,g) with strong bounded geometry, as in Definition 2.3, involving the scalar curvature function. Namely in strong bounded geometry, if the supremum of scalar curvature function Sg<n(n−1)k0 for some $k_…
There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation f(D2u)=0. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman …
Sharp generalization of boundary regularity for area minimizing currents with arbitrary multiplicity.
problem Boundary regularity of area minimizing currents with multiplicity.
method Sharp generalization of Allard's boundary regularity theorem to higher multiplicity settings.
result The set of density Q/2 singular boundary points of T is Hm−3-rectifiable. We prove that every continuous function on a separable infinite-dimensional Hilbert space X can be uniformly approximated by smooth functions with no critical points. This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as …
In this paper, we investigate Liu-Xu-Ye-Zhao's conjecture [30] and prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in spheres which improves the convergence theorem of Baker [2] as well as the differentiable sphere theorems of Gu-Xu-Zhao [16, 50, 52].
Strong Frankel theorem for shrinkers in all dimensions.
problem Intersection of shrinkers in large balls.
method Proof using strong Bernstein theorem for stable Gaussian surfaces.
result Shrinkers are connected in all large balls.
Sharp convergence theorem for Yang-Mills flow on ALE manifolds proved.
problem Proving convergence of Yang-Mills flow on ALE gravitational instantons.
method Noncompact version of the 'parabolic gap theorem'.
result Sharp convergence theorem for Yang-Mills flow on ALE 4-manifolds.
We prove that every continuous mapping from a separable infinite-dimensional Hilbert space X into Rm can be uniformly approximated by C∞ smooth mappings {\em with no critical points}. This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some…
Sharp heat equation gradient estimates on compact manifolds.
problem Gradient estimates for positive solutions on weighted manifolds.
method Proving sharp gradient estimates for positive solutions to the weighted heat equation.
result Refined gradient estimates and Liouville theorems for ancient solutions.
Study Liouville theorems for harmonic maps along ancient super Ricci flows.
problem Proving Liouville theorems for harmonic maps under specific geometric conditions.
method Using Perelman's reduced geometric viewpoint, derive Liouville theorems with controlled growth.
result Sharp growth conditions and new Liouville theorems for both non-positively and positively curved target spaces.
We refine Theorem A due to Gursky \cite{G3}. As applications, we give some rigidity theorems on four-manifolds with postive Yamabe constant. In particular, these rigidity theorems are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of e…
Sharp bounds on scalar curvature spectrum and rigidity theorems.
problem Understanding scalar curvature bounds and rigidity on manifolds.
method Sharp upper bounds for the bottom spectrum of the Beltrami Laplacian, scalar curvature rigidity theorem.
result Sharp upper bound for the bottom spectrum of the Beltrami Laplacian and scalar curvature rigidity theorem.
Motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS spacetimes, we initiate the study of massive scalar waves satisfying □gψ−μψ=0 on the interior of Anti-de Sitter (AdS) black holes. We prescribe initial data on a spacelike hypersurface of a Reissner--Nordström--AdS black hole and impose…
Sharp comparison theorems for 3D manifolds with scalar curvature bound.
problem Understanding the geometry and topology of 3D manifolds with scalar curvature constraints.
method Sharp comparison results for Green's function and spectrum, derived from scalar curvature bounds.
result Sharp upper and lower bounds for the Green's function and spectrum of 3D manifolds.
Defines coupled embeddability for maps on products of spaces, generating examples and nonexamples.
problem Understanding when maps on products of spaces can be embedded.
method Uses known results for nonsingular biskew and bilinear maps, studies genericity properties, extends Whitney embedding theorems, and relates to Z/2-coindex of embedding spaces. result Generates strong obstructions to coupled embeddability in terms of combinatorics of triangulations.
We show that a closed, connected and orientable Riemannian manifold of dimension d that admits a quasiregular mapping from Rd must have bounded cohomological dimension independent of the distortion of the map. The dimension of the degree l de Rham cohomology of M is bounded above by (ld). Thi…
Sharp spectral theorems and isoperimetric inequalities for manifolds with nonnegative Ricci curvature.
problem Understanding the geometry and topology of manifolds with nonnegative Ricci curvature.
method New spectral inequalities and isoperimetric problems involving unequal weights and warped bubbles.
result Sharp spectral and isoperimetric bounds for manifolds with nonnegative Ricci curvature.
The paper proves Sard's theorem for polynomial maps in infinite dimensions.
problem The validity of Sard's theorem for polynomial maps in infinite-dimensional Banach manifolds.
method Sharp quantitative criteria for the validity of Sard's theorem.
result The paper provides criteria for the validity of Sard's theorem in infinite-dimensional Banach manifolds.
New extension theorem for projective manifolds.
problem Extension of twisted canonical forms on hypersurfaces.
method Established a new extension result for canonical forms on hypersurfaces with simple normal crossings.
result Obtained sharp bounds for the extension.
We conduct a post hoc analysis of solar flare predictions made by a Long Short Term Memory (LSTM) model employing data in the form of Space-weather HMI Active Region Patches (SHARP) parameters calculated from data in proximity to the magnetic polarity inversion line where the flares originate. We train the the LSTM mod…