Study proves boundedness of operators in variable exponent Morrey spaces.
problem Boundedness of operators in global Morrey-type spaces with variable exponents.
method Analysis of Hardy-Littlewood maximal operator and potential type operator in variable exponent Morrey spaces.
result Boundedness of the Hardy-Littlewood maximal operator and potential type operator in global Morrey-type spaces with variable exponents.
We provide a simpler proof and slight strengthening of Morrey's famous lemma on ε-conformal mappings. Our result more generally applies to Sobolev maps with values in a complete metric space and we obtain applications to the existence of area minimizing surfaces of higher genus in metric spaces. Unlike Morr…
Study short-time existence of Ricci-DeTurck flow from rough metrics with Morrey-type integrability.
problem Short-time existence of Ricci-DeTurck flow from rough metrics with specific integrability condition.
method Rough existence theory, preservation and improvement of scalar curvature bounds.
result Preservation and improvement of distributional scalar curvature lower bounds under certain conditions.
The paper proves parabolic gap theorems for Yang-Mills energy.
problem Yang-Mills energy and instantons on various manifolds.
method Parabolic Yang-Mills flow and Morrey norms.
result Spaces of connections with Yang-Mills energy less than a certain threshold deformation-retract onto spaces of instantons.
Develops topological concepts for Morrey-Sobolev bundles in high dimensions.
problem Lack of continuity in transition maps for Morrey-Sobolev bundles.
method Introduces topological isomorphism classes and uses connection-oriented approach.
result Derives approximability results for bundles and connections in Morrey-Sobolev setting.
Generalizes Ricci flow starting from small curvature concentration with a Morrey-type condition.
problem Ricci flow starting from manifolds with unbounded curvature.
method Replaces bounded curvature with a Morrey-type condition on the gradient of the metric relative to a complete bounded curvature metric.
result Long-time existence of Ricci flow with curvature decay estimates and diffeomorphic manifold.
In four and higher dimensions, we show that any stationary admissible Yang-Mills field can be gauge transformed to a smooth field if the L2 norm of the curvature is sufficiently small. There are three main ingredients. The first is Price's monotonicity formula, which allows us to assert that the curvature is small n…
Sharp inequalities and eigenvalue problems on Finsler manifolds with nonnegative Ricci curvature.
problem Establishing sharp Morrey-Sobolev inequalities and eigenvalue problems on Finsler manifolds.
method Combining sharp isoperimetric inequality and anisotropic symmetrization argument.
result Existence and multiplicity of solutions for eigenvalue problems and elliptic PDEs.
The paper improves energy decay estimates for Dir-stationary Q-valued functions and applies them to Liouville-type theorems and continuity.
problem Improving energy decay estimates for Dir-stationary Q-valued functions.
method Establishing improved decay estimates and applying them to derive Liouville-type theorems and continuity.
result Dir-stationary Q-valued functions exhibit the Lebesgue property and reside in a generalized Campanato-Morrey space.
We give new estimates for a critical elliptic system introduced by Rivière-Struwe in \cite{riviere_struwe} (see also the work of Rupflin \cite{rupflin} and Schikorra \cite{schikorra_frames}), which generalises PDE solved by harmonic (and almost harmonic) maps from a Euclidean ball $B_1 \In \R^n$ into Riemannian manifol…
The paper estimates solutions to a heat inequality on Riemannian manifolds with specific initial data.
problem Estimating nonnegative solutions to a semilinear heat inequality with Morrey norms.
method Using differential inequalities and Morrey norms, the paper obtains L∞ estimates and improved estimates near the initial time. result Improved estimates for nonnegative solutions of the differential inequality in Morrey norms on Riemannian manifolds.
The study shows how to regularize weakly harmonic maps using Sobolev norms and Coulomb frames.
problem Regularity of weakly harmonic maps between Riemannian manifolds.
method New structure equations and Coulomb-frame methods combined with Hardy-BMO duality.
result Sufficient conditions on Sobolev norms ensure full regularity of weakly harmonic maps.
Researchers extend regularity of p-harmonic maps into spheres for a new range of p.
problem Establishing regularity of p-harmonic maps for a broader range of p. method Combining Morrey's methods with Hardt and Lin's Extension Theorem, and proving a sharp Kato inequality.
result Regularity for p∈[2.961,3] and p∈[2,p0] with p0≈2.366. We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadra…
Given two Jordan curves in a Riemannian manifold, a minimal surface of annulus type bounded by these curves is described as the harmonic extension of a critical point of some functional (the Dirichlet integral) in a certain space of boundary parametrizations. The H2,2-regularity of the minimal surface of annulus t…
We give a new proof of the existence (\cite{HM}, \cite{Ren}) of a Jenkins-Strebel differential Φ on a Riemann surface $\SR$ with prescribed heights of cylinders by considering the harmonic map from $\SR$ to the leaf space of the vertical foliation of Φ, thought of as a Riemannian graph. The novelty of the argument …
The paper proves a decomposition theorem for forms on sub-Riemannian contact manifolds.
problem Developing a Lp-Hodge decomposition on sub-Riemannian contact manifolds. method Using a Sobolev approach and recent results from [4] and [6].
result Established an Lp-Hodge decomposition theorem for Rumin's forms on sub-Riemannian contact manifolds. We study the regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler--Lagrange equations and consider the regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under…
The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…
This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine Lp−Sobolev inequalities. The logarithmic version of affine Lp−Sobolev inequalities is verified. Moreover, An alternative proof of the affine Mo…
The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchhei…
We prove a Morrey-type theorem for Hamiltonian stationary submanifolds of Cn. Namely, if L ⊂ Cn is a C1 Lagrangian submanifold with weakly harmonic Lagrangian phase θ, then L must be smooth. In the process we also discuss a local version of the equation, which is a nonline…
The study establishes bounds for Schrödinger operators on Riemannian manifolds.
problem Bounding Schrödinger operators on Riemannian manifolds.
method Utilizes weighted manifolds and Faber-Krahn inequalities to derive bounds.
result Establishes conditions for Schrödinger operators to be positive and for their spectra.
The paper develops L2 theory for foliations on manifolds with boundary.
problem Analyzing the cohomology of foliated manifolds with boundary.
method Develops L2 theory, establishes decomposition and vanishing theorems, and proves duality and extension theorems. result Establishes Dolbeault decomposition of basic forms and proves global regularity for ∂ˉB-equations. We show the existence of Yang--Mills--Higgs (YMH) fields over a Riemann surface with boundary where a free boundary condition is imposed on the section and a Neumann boundary condition on the connection. In technical terms, we study the convergence and blow-up behavior of a sequence of Sacks-Uhlenbeck type α-YMH fiel…
We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension n≥5. We define the natural function space A_{G} in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A_{G} can be also…
For the class of approximate harmonic maps u∈W1,2(Σ,N) from a closed Riemmanian surface (Σ,g) to a compact Riemannian manifold (N,h), we show that (i) the so-called energy identity holds for weakly convergent approximate harmonic maps {un}:Σ→N, with tension fields τ(un) bounded in the Morrey spa…
Symmetric hypersurfaces and boundaries in R^n+1 with group actions.
problem Symmetry of hypersurfaces with symmetric boundaries.
method Infinitesimal Lie group actions, Cauchy problem, Morrey's regularity theory, Cauchy-Kovalevskaya Theorem.
result Symmetry inheritance for minimal and CMC hypersurfaces with symmetric boundaries.
Smooth Yang-Mills fields proved in supercritical dimensions.
problem Regularity of weak Yang-Mills connections in high dimensions.
method ε-regularity theorem, Coulomb gauges construction.
result Stationary Yang-Mills fields are smooth away from a small singular set.
Extending isometric immersions with low regularity, especially supercritical.
problem Finding isometric immersions with low regularity in Euclidean space.
method Utilising Uhlenbeck gauges and compensated compactness theory.
result Existence of isometric immersions with low regularity, including supercritical cases.
The paper converts metric bounds to distance function Hölder bounds and proves compactness theorems.
problem Proving geometric stability results with scalar curvature bounds.
method Transforming Lp bounds to Hölder bounds for distance functions. result Compactness theorems and convergence guarantees for Riemannian manifolds.
Introduces a new Hodge theory using vector fields on manifolds.
problem Developing a new Hodge theory for manifolds with vector fields.
method Defines a vector field induced Hodge L2-inner product, codifferential, and Laplacian. result Established de Rham-Hodge theory for closed and boundary manifolds.
We consider a compact, oriented, smooth Riemannian manifold M (with or without boundary) and we suppose G is a torus acting by isometries on M. Given X in the Lie algebra and corresponding vector field XM on M, one defines Witten's inhomogeneous coboundary operator dXM=d+ιXM:ΩG±→ΩG∓…
We consider geometric and analytical aspects of M-theory on a manifold with boundary Y. The partition function of the C-field requires summing over harmonic forms. When Y is closed Hodge theory gives a unique harmonic form in each de Rham cohomology class, while in the presence of a boundary the Hodge-Morrey-Friedrichs…
Establishes interior regularity results for a broad class of two-dimensional nonlinear elliptic systems using a unified abstract framework.
problem Interior regularity results for two-dimensional nonlinear elliptic systems
method A unified abstract framework built around a Campanato-type discrete iteration scheme coupled with a Caccioppoli-type estimate
result Local Hölder continuity of the map u is established, with an explicit Hölder exponent that optimally attains the classical Morrey--Campanato threshold dictated by the Lebesgue integrability of the source term f Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
(1,1) non-L-space knots are foliar in 3D space.
problem Proving (1,1) non-L-space knots are foliar.
method Analyzing (1,1) non-L-space knots in S3 and lens spaces. result (1,1) non-L-space knots are persistently foliar.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
The distance function ϱ(p,q) (or d(p,q)) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over Rn, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
In a paper (math.DG/0403528) we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, …
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.