Smooth submetries between curved spaces are smooth.
problem Smoothness of submetries between curved spaces.
method Proving smoothness of submetries in a general setting, including Riemannian submersions and isometric actions.
result Smoothness of the base manifold is implied by the smoothness of the total manifold without curvature assumptions.
Smooth manifolds from locally homogeneous spaces.
problem Understanding smoothness in C0-Riemannian manifolds. method Demonstrated locally homogeneous C0-Riemannian manifolds are smooth. result Locally homogeneous C0-Riemannian manifolds are smooth. Smooth Riemannian manifolds can be embedded without boundary.
problem Embedding smooth Riemannian manifolds with boundary into complete manifolds without boundary.
method General gluing and conformal-deformation construction.
result Any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannian manifold without boundary.
Study shows not all smooth paths are optimal in certain geometric structures.
problem Existence of non-smooth sub-Riemannian minimizing geodesics.
method Constructed a C2 but not C3 length-minimizer example. result Found a real-analytic sub-Riemannian structure with non-smooth minimizers.
Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.
problem Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
method Riemannian geometry of Bergman metrics and smoothness of families of isometries.
result Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
Riemannian metrics and Laplacians defined for complex distributions on manifolds.
problem Defining metrics and Laplacians for distributions on manifolds of varying rank.
method Introduced a Riemannian metric and Laplace operator for generalised smooth distributions on manifolds.
result Essentially self-adjoint Laplacian on compact manifolds, hypoellipticity proven.
We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.
Smooth projections on manifolds split into simpler parts.
problem Decomposing operators on Riemannian manifolds.
method Constructing a sum of smooth orthogonal projections.
result Extends decomposition on real line to manifolds.
Smooth solutions found for modified mean curvature flow in Riemannian manifolds.
problem Existence of smooth solutions for modified mean curvature flow.
method A priori estimates for modified mean curvature flow in Riemannian manifolds with Killing vector field.
result Existence of smooth, entire, longtime solutions for modified mean curvature flow with smooth initial data.
Proof shows smooth minimal hypersurfaces for perimeter-minimizing sets in low-dimensional Riemannian manifolds.
problem Finding sets of least perimeter in Riemannian manifolds.
method Short proof using de Giorgi and Miranda's tradition in flat space.
result Reduced boundary of least perimeter sets is a smooth minimal hypersurface in low dimensions.
Let Mn be a complete, non-compact and C∞-smooth Riemannian manifold with nonnegative sectional curvature. Suppose $\Cal S$ is a soul of Mn. Then any distance non-increasing retraction $Ψ: M^n \to \Cal S$ must give rise to a C∞-smooth Riemannian submersion.
The study extends calibrated geometry to smooth maps and finds energy bounds.
problem Finding energy bounds for smooth maps between Riemannian manifolds.
method Generalizing calibrated submanifolds to smooth maps and applying to energy functional.
result Lower bounds to the energy of smooth maps in homotopy classes.
Let M be a differentiable manifold. We say that a tensor field g defined on M is non-regular if g is in some local Lp space or if g is continuous. In this work we define a mollifier smoothing g_t of g that has the following feature: If g is a Riemannian metric of class C2, then the Levi-Civita connection and the Rieman…
Smooth metrics satisfying Penrose inequality are necessarily smooth.
problem Rigidity of Penrose inequality with singular metrics.
method Showed suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality are smooth in specified coordinates.
result Smooth metrics satisfying Penrose inequality are necessarily smooth.
We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the 1-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices…
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
We begin by showing that every real analytic orbifold has a real analytic Riemannian metric. It follows that every reduced real analytic orbifold can be expressed as a quotient of a real analytic manifold by a real analytic almost free action of a compact Lie group. We then extend a well-known result of Nomizu and Ozek…
Survey on smooth function and form density in Riemannian Sobolev spaces.
problem Density of smooth functions and forms in Sobolev spaces on Riemannian manifolds.
method Careful examination of weak covariant derivatives and partial derivatives.
result Equivalence of weak covariant derivatives to weak partial derivatives.
Extends heat flow estimates to non-smooth spaces.
problem Heat flow estimates on non-smooth metric measure spaces.
method Extends Hamilton's gradient estimates and monotonicity formula to metric measure spaces.
result Establishes heat flow estimates for metric measure spaces.
The study proves smoothness and estimates for p-harmonic mappings between Riemannian manifolds.
problem Smoothness and estimates for p-harmonic mappings between Riemannian manifolds. method Analyzing stationary and minimizing p-harmonic mappings with specific curvature conditions. result Smoothness and estimates for p-harmonic mappings under certain curvature conditions. Let M be a Riemannian manifold with a polar action by the Lie group G, with section Σ⊂M and generalized Weyl group W. We show that restriction to Σ is a surjective map from the set of smooth G-invariant tensors on M onto the set of smooth W-invariant tensors on Σ. Moreover, we show that every s…
Smooth approximation of integral cycles mod 2 in Riemannian manifolds.
problem Approximating mod 2 integral cycles by smooth submanifolds.
method Approximation of mod 2 integral cycles by smooth submanifolds with controlled singularities.
result Every mod 2 integral cycle can be approximated by a smooth submanifold with a controlled singular set.
Smooths metrics with nonnegative scalar curvature near singular sets.
problem Approximating metrics with nonnegative scalar curvature near singularities.
method Ricci-DeTurck flow to approximate metrics.
result Approximated metrics converge to the original metric in C∞ away from the singular set. Smooth bundles with rough data maintain Hodge kernel isomorphism.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
New obstruction found for smoothability of certain 4-manifolds.
problem Obstructing the smoothability of Riemannian metrics with non-positive sectional curvature.
method Extending Davis-Januszkiewicz-Lafont methods to construct examples of locally CAT(0) 4-manifolds with specific properties.
result Examples of locally CAT(0) 4-manifolds that do not have a Riemannian smoothing despite satisfying isolated flats condition.
The paper tackles the smooth Riemannian extension problem, providing existence results and obstructions.
problem Addressing the problem of extending a Riemannian manifold with smooth boundary to a geodesically complete one.
method Providing three types of results: existence theorems, topological obstructions, and existence results under certain conditions.
result Existence of geodesically complete Riemannian extensions without curvature constraints, various obstructions, and specific existence results under convexity conditions.
We prove here that given a proper isometric action K×M→M on a complete Riemannian manifold M then every continuous isometric flow on the orbit space M/K is smooth, i.e., it is the projection of an K-equivariant smooth flow on the manifold M. As a direct corollary we infer the smoothness of isometric …
Riemannian gradient descent helps escape saddle points on curved spaces.
problem Minimizing nonconvex functions on curved spaces (Riemannian manifolds).
method Perturbed Riemannian gradient descent algorithm.
result Converges to second-order stationary points, matching unconstrained smooth minimization rates.
We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous ε:M→(0,+∞), and for every positive number r>0, there exists a C∞ smooth Lipschitz function g:M→R such that ∣f(p)−g(p)∣≤ε(p) for every …
Generic smooth boundaries for isoperimetric regions in 8D manifolds.
problem Understanding boundaries of isoperimetric regions in high-dimensional spaces.
method Generic regularity results for isoperimetric regions in closed Riemannian manifolds of dimension eight.
result Smooth nondegenerate boundaries for isoperimetric regions for generic metrics and volumes.
We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.
The paper investigates exotic smooth structures on manifolds with group actions.
problem Existence of homeomorphic but not diffeomorphic smooth manifolds with shared basic spectra.
method Investigates Riemannian Laplacian eigenvalues and eigenfunctions on manifolds with compact Lie group actions.
result Establishes the existence of homeomorphic yet not diffeomorphic manifolds with shared basic spectra.
Researchers define geodesic curvature for 3D sub-Riemannian curves, improving distance calculations.
problem Improving sub-Riemannian distance calculations for 3D curves.
method Introducing geodesic curvature kζ for smooth horizontal curves in 3D contact sub-Riemannian manifolds. result Geodesic curvature appears as the first corrective term in the Taylor expansion of sub-Riemannian distance.
Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectr…
We smooth the singularities of a strictly hyperbolized smooth cube manifold.
Smooth approximation of integral cycles in manifolds.
problem Approximating integral cycles in Riemannian manifolds.
method Approximation of integral cycles by smooth submanifolds with controlled area and singularities.
result Integral cycles can be approximated by smooth submanifolds with controlled area and singularities.
The study proves the stability of smooth embeddings of Riemannian metrics into Euclidean space.
problem Stability of smooth embeddings of Riemannian metrics into Euclidean space.
method Local perturbation method to derive a time-dependent local perturbation method.
result Construction of a smooth parametrized family of isometric embeddings for a short time.
Real analytic submersion images are always real analytic.
problem Analyticity of submersion images
method Analyticity proof for Riemannian submersions
result Image of real analytic submersion is real analytic
We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds.
Study shows area-minimizing submanifolds are mostly rough, not smooth.
problem Understanding the smoothness of area-minimizing submanifolds.
method Proved non-smoothness by contradiction and established Hausdorff dimension bounds.
result Area-minimizing submanifolds are not generically smooth, resolving a conjecture.
Develops calculus on Wasserstein spaces for Riemannian manifolds.
problem Characterizing and understanding the geometry of Wasserstein spaces.
method Intrinsic formalism for topology, smooth structure, and Riemannian geometry of Wasserstein spaces.
result Wasserstein spaces of closed manifolds are geodesically convex.
Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be smoothed to having a sectional curvature bound. This partly extends previous a priori estimates of Ye Li (J. Geom. Anal. 17 (20…
Study rough Riemannian metrics on manifolds, proving their connectedness and completeness.
problem Understanding the space of all locally elliptic and bounded Riemannian metrics on manifolds.
method Introduced an extended metric space and proved its properties.
result Proved the space of rough Riemannian metrics is complete and connected.
Researchers create smooth metrics with specific curvature properties.
problem Creating smooth metrics with desired curvature properties.
method Carlotto-Schoen-type gluing on cone-like sets.
result Smooth metrics with specific scalar curvature can be created.
Geodesic completeness for Riemannian metrics on smooth probability densities is studied.
problem None of the studied Riemannian metrics are geodesically complete.
method Analysis of Hamilton--Jacobi-like partial differential equations, providing order conditions for global existence and uniqueness.
result Geodesic completeness is established for a class of higher-order Sobolev type metrics.
Locally homogeneous RCD spaces are shown to be smooth manifolds.
problem Understanding the structure of RCD spaces.
method Adapting existing results to new spaces.
result Locally homogeneous RCD spaces are isometric to smooth manifolds.
The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.
problem Boundary estimates for fully nonlinear Yamabe equations on Riemannian manifolds.
method Deriving a priori second derivative estimates for subsolutions.
result Existence of smooth solutions with uniform estimates.
Proves Slice Theorem for singular Riemannian foliations, generalizing Schwarz results.
problem Understanding algebra of smooth basic functions in singular Riemannian foliations.
method Proves Slice Theorem and uses it to study C∞-algebra of smooth basic functions. result Generates algebra by a finite number of polynomials in infinitesimal case.