Optimizes shapes on non-standard manifolds.
problem Optimization on non-standard infinite-dimensional manifolds.
method Develops gradient descent on weak Riemannian manifolds.
result Establishes foundational properties for optimization on various weak Riemannian manifolds.
Defines weak geodesics on specific subsets of manifolds.
problem Characterizing geodesics on prox-regular subsets of Riemannian manifolds.
method Defining weak geodesics as continuous curves with weak regularities, and characterizing them as viscosity critical points of the energy functional.
result Characterizes weak geodesics on prox-regular subsets of Riemannian manifolds.
The paper proves properties of curves in Riemannian manifolds.
problem Characterizing curves in Riemannian manifolds.
method Analyzing locally minimizing and weak geodesics.
result Locally minimizing curves are weak geodesics under certain conditions.
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.
The paper explores generalized Riemannian manifolds with weak metric structures.
problem Understanding weak metric structures on generalized Riemannian manifolds.
method Study of generalized metric connections, weak metric structures, and skew-symmetric torsion.
result Generalized Riemannian manifolds can be split into nearly Kähler manifolds.
We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isom…
Gradient estimates for solutions to a p-Laplacian equation on Riemannian manifolds.
problem Gradient estimates for positive weak solutions to a p-Laplacian equation on Riemannian manifolds.
method Morser iteration technique
result Gradient estimates show that positive weak solutions do not exist under certain conditions on manifolds with nonnegative Ricci curvature.
The study examines conditions for weak nearly cosymplectic manifolds to split into products.
problem Understanding the curvature and topology of weak nearly cosymplectic manifolds.
method Analyzes the conditions for splitting and characterizes specific manifolds.
result Conditions for weak nearly cosymplectic manifolds to become Riemannian products are identified.
We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian m…
Extends weak continuity of Yang-Mills connections to a broader class.
problem Weak compactness of Ω-Yang-Mills connections. method Compensation compactness argument applied to Yang-Mills fields.
result Weak continuity result extended to Ω-Yang-Mills connections. Proves curvature tensor convergence for smoothable spaces.
problem Curvature tensor behavior in smoothable Alexandrov spaces.
method Weak convergence of curvature tensors in noncollapsing sequences.
result Proves convergence of curvature tensors in smoothable Alexandrov spaces.
Study on a new type of manifolds that generalize almost C-manifolds.
problem Understanding weak nearly C-manifolds and their properties.
method Analyzing conditions for local Riemannian product structures and characterizing specific dimensions.
result Conditions for a weak nearly C-manifold to become locally a Riemannian product and characterization of specific dimensions.
We prove that the L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L^2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponentia…
We prove the Focal Index Lemma and the Rauch and Berger comparison theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop spaces of a compact manifold with the L^2 metrics
Introduces a new framework for Riemannian diffeology.
problem No specific problem stated; focuses on a new framework.
method Uses tangent functor and metric from Iglesias-Zemmour to establish weak Riemannian diffeological spaces.
result Establishes a category of weak Riemannian diffeological spaces and shows induced pseudodistance is a distance under technical conditions.
Survey of recent results in weak almost contact structures.
problem New geometric structures replacing complex structure in contact manifolds.
method Survey of recent findings in weak almost contact manifolds.
result Recent results on geodesic and Killing fields, rigidity and splitting theorems, etc.
Sharp upper bounds found for solutions of a specific equation on Riemannian manifolds.
problem Finding upper bounds for solutions of a specific equation on Riemannian manifolds.
method Proved sharp upper estimates of weak subsolutions to the Leibenson equation on Riemannian manifolds with non-negative Ricci curvature.
result Improved and proved a conjecture about upper bounds for solutions of the Leibenson equation.
We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is ob…
Quantizes Willmore energy in Riemannian manifolds with bounded energy and area.
problem Quantization of Willmore energy in bounded energy and area conditions.
method Uniform boundedness of Willmore energy and area, weak convergence of maps, and conformal structures in compact domain.
result Quantization of Willmore energy holds under specified conditions.
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.
problem Bounding weak subsolutions of Trudinger's equation on Riemannian manifolds.
method Proving sub-Gaussian upper bounds for weak subsolutions.
result The upper bounds are sharp for specific classes of manifolds, including \(\mathbb{R}^{n}\).
The paper explores new connections in non-symmetrical gravitational theory using weak metric structures.
problem Investigating linear connections in non-symmetrical gravitational models.
method Study of generalized Riemannian manifolds with weak metric structures and connections satisfying specific conditions.
result Proved properties of connections and structures in generalized Riemannian manifolds.
The study explores new metric structures on manifolds, linking them to Einstein metrics.
problem Characterizing and understanding weak K-contact manifolds and their properties.
method Analyzing weak K-contact manifolds and their properties, including the parallel Ricci tensor and generalized Ricci soliton structures.
result Sufficient conditions for weak K-contact manifolds with specific properties to be Einstein manifolds.
The study proves a theorem on Riemannian manifolds for wedge products of weakly convergent differential forms.
problem Analyzing the limiting behavior of wedge products of weakly convergent differential forms on Riemannian manifolds.
method Formulating and proving compensated compactness theorems for wedge products of differential forms on closed Riemannian manifolds.
result The theorem generalizes the div-curl lemma for vectorfields and applies to critical regularity exponents.
The system of weak normality equations constitutes a part in the complete system of normality equations. Solutions of each of these two systems of equations are associated with some definite classes of Newtonian dynamical systems in Riemannian manifolds. In this paper for the case of simplest flat Riemannian manifold $…
Proves existence of solutions for a specific nonlinear equation on Riemannian manifolds.
problem Existence of solutions for a doubly nonlinear evolution equation on Riemannian manifolds.
method Proves existence of weak solutions using the Leibenson equation.
result Proves the existence of a unique weak solution for any initial condition in L1(M)∩L∞(M). Study proves upper bounds for solutions on Riemannian manifolds.
problem Proving upper bounds for solutions of Leibenson's equation on Riemannian manifolds.
method Proved upper bounds equivalent to a euclidean-type Sobolev inequality.
result Upper bounds for solutions of Leibenson's equation on Riemannian manifolds are equivalent to euclidean-type Sobolev inequalities.
We prove that SU(n) (n > 2) and Sp(n)U(1) (n > 1) are the only connected Lie groups acting transitively and effectively on some sphere which can be weak holonomy groups of a Riemannian manifold without having to contain its holonomy group. In both cases the manifold is Kaehler.
Study optimal transport on globally hyperbolic spacetimes, focusing on weak Kantorovich potentials' regularity.
problem Investigate regularity of weak Kantorovich potentials on globally hyperbolic spacetimes.
method Apply insights from Riemannian and Lorentzian cases to study π-solutions. result Conclude existence, uniqueness, and structure of optimal transport maps.
We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…
There is a well developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed are homogeneity, geodesic completeness, the geodesic orbit…
Newtonian dynamical systems which accept the normal shift on an arbitrary Riemannian manifold are considered. For them the determinating equations making the weak normality condition are derived. The expansion for the algebra of tensor fields is constructed.
The paper proves no positive weak solutions for a specific inequality on Riemannian manifolds.
problem Proving the nonexistence of positive weak solutions for a quasilinear inequality on Riemannian manifolds.
method Analyzing the volume growth of geodesic balls and applying Liouville-type theorems.
result The inequality has no positive weak solutions under certain volume growth conditions.
Study weak f-K-contact manifolds, finding Einstein-type metrics and solitons.
problem Characterize and study geometric properties of weak f-K-contact manifolds. method Analyzing weak metric f-structures, using Killing vector fields, and Jacobi operators. result Einstein weak f-K-contact manifolds are Ricci flat. This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the L2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold. The main body of the thesis is a description of the completion manifold of metrics with respect to the $L^…
The paper presents the Einstein connection for nonsymmetric pseudo-Riemannian manifolds.
problem The challenge is to define the Einstein connection for nonsymmetric pseudo-Riemannian manifolds.
method The approach involves using a weak almost contact structure and a linear connection with torsion.
result Explicit formulas for the Einstein connection are provided.
In this paper we generalize the Local Removable Singularity Theorem in [16] for minimal laminations to the case of weak H-laminations (with H∈R constant) in a punctured ball of a Riemannian three-manifold. We also obtain a curvature estimate for any weak CMC foliation (with possibly varying constant mea…
On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harm…
Extends mapping results to non-compact Riemannian manifolds with positive reach.
problem Extending mapping results to non-compact Riemannian manifolds with positive reach.
method Using a criterion by A. Petrunin and results by B. Bulanyi and J. Van Schaftingen, the study extends critical Sobolev mappings.
result Extended maps satisfy an exponential weak-type Sobolev-Marcinkiewicz estimate.
Preserves scalar curvature bounds under weak convergence of 3-manifolds.
problem Preserving scalar curvature bounds under weak convergence of 3-manifolds.
method Comparison between μ-bubbles in M_k and M.
result Scalar curvature lower bounds are preserved under weak convergence.
Finite time for subsolutions on Riemannian manifolds proved.
problem Finite extinction time for subsolutions of a specific equation on Riemannian manifolds.
method Proved finite extinction time using weighted Sobolev inequality and assumptions on p, q, and ρ.
result Weak subsolutions to the equation have a finite extinction time.
Study on Kähler manifolds proves weak decompositions and relates harmonic forms.
problem Analyzing harmonic forms on Kähler manifolds.
method Proves weak W1,2 Bott-Chern and Dolbeault decompositions. result Strict relation between W1,2 Bott-Chern harmonic forms and the W1,2 Bott-Chern decomposition. Study on self-similar sets on Riemannian manifolds with new separation conditions.
problem Analyzing self-similar sets on Riemannian manifolds with new separation conditions.
method Formulated weak separation and finite type conditions for conformal iterated function systems on Riemannian manifolds.
result Obtained formulas for Hausdorff dimensions of self-similar and graph self-similar sets.
Study on potential behavior in special geometric spaces.
problem Understanding potential behavior in specific geometric spaces.
method Analyzing asymptotic behavior of p-capacitary potentials and weak Inverse Mean Curvature Flow. result Characterized the behavior of potentials in Asymptotically Conical manifolds.
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.
We observe that a vanishing geodesic distance arising from a weak Riemannian metric in a Hilbert manifold can be constructed.
We study weakened f-structures on manifolds, generalizing classical results.
problem Classical f-structures and their properties on manifolds. method Introduced and studied weakened f-structures, subclasses, and their properties. result Generalized known results on globally framed f-manifolds. In infinite dimensional Heisenberg group, degenerate distances linked to unbounded curvature.
problem Degenerate distances and unbounded curvature in infinite dimensional Heisenberg group.
method Construct left invariant weak Riemannian and sub-Riemannian metrics, adapt sectional curvature definition.
result Degenerate distances coincide with unbounded sectional curvature.
We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on negatively curved Riemannian manifolds. We show existence of solutions taking as initial condition a finite Radon measure, not necessarily positive. We then establish uniqueness in the class of nonnegative …