Curvature tensor of smoothable Alexandrov spaces
We prove weak convergence of curvature tensors of Riemannian manifolds for converging noncollapsing sequences with a lower bound on sectional curvature.
math.DG↗
arXiv subjects
Anton Petrunin
Publications and source records attributed to Anton Petrunin.
Metric minimizing surfaces revisited
A surface which does not admit a length nonincreasing deformation is called metric minimizing. We show that metric minimizing surfaces in CAT(0) spaces are locally CAT(0) with respect to their intrinsic metric.
math.DG↗
Extra pearls in graph theory
This is a supplement for "Pearls in graph theory" -- a textbook written by Nora Hartsfield and Gerhard Ringel. We discuss the following topics: Ramsey theory, Probabilistic method, Deletion-contraction formulas, Matrix theorem, Graph-polynomials, Generating functions, Minimum spanning trees, Marriage theorem and its relatives, Toroidal graphs, Rado graph, Rewriting systems.
math.CO↗
Monotonicity of saddle maps
We prove an analog of the Schoen-Yau univalentness theorem for saddle maps between discs.
math.DG↗
Alexandrov geometry: foundations
Alexandrov spaces are defined via axioms similar to those given by Euclid. The Alexandrov axioms replace certain equalities with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded above and curvature bounded below. The definitions of the two classes of spaces are similar, but their properties and known applications are quite different. Our approach is novel in its attention to the interrelatedness of the two fields, and its emphasis on the way each illuminates the other. The goal of this book is to give a comprehensive exposition of the structure theory of Alexandrov spaces with curvature bounded above and below. It includes all the basic material as well as selected topics inspired by considering the two contexts simultaneously. We only consider the intrinsic theory, leaving applications aside. This book includes material up to the definition of dimension. Another volume still in preparation will cover further topics.
math.DG↗
All-set-homogeneous spaces
A metric space is said to be all-set-homogeneous if any of its partial isometries can be extended to a genuine isometry. We give a classification of a certain subclass of all-set-homogeneous length spaces.
math.MG↗
Graph comparison meets Alexandrov
Graph comparison is a certain type of condition on metric space encoded by a finite graph. We show that any nontrivial graph comparison implies one of Alexandrov's comparisons. The proof gives a complete description of graphs with trivial graph comparisons.
math.MG↗
Pure metric geometry: introductory lectures
We discuss domestic affairs of metric spaces, keeping away from any extra structure. Topics include universal spaces, injective spaces, Hausdorff and Gromov--Hausdorff convergences, and ultralimits.
math.MG↗
Five-point Toponogov theorem
We give an if-and-only-if condition on a five-point metric spaces that admit embeddings into nonnegatively curved Riemannian manifolds.
math.DG↗
Alexandrov's embedding theorem
Written for the book "Mathematicians from Saint Petersburg and their theorems".
math.HO↗
Polyhedral approximations of Riemannian manifolds
We give a condition on the curvature tensors of Riemannian manifolds that admit Lipschitz approximation by polyhedral metrics with curvature bounded below or above. We show that this condition is also sufficient for the existence of local approximations. We conjecture that it is also sufficient for the global approximations and prove it in some special cases.
math.DG↗
Weak topology on CAT(0) spaces
We analyze weak convergence on CAT(0) spaces and the existence and properties of corresponding weak topologies.
math.MG↗
5-point CAT(0) spaces after Tetsu Toyoda
We give another proof of Toyoda's theorem that describes 5-point subpaces in CAT(0) length spaces
math.MG↗
Cyclic projections in Hadamard spaces
We show that cyclic products of projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bačák. Namely, we construct an example of convex subsets in a Hadamard space such that the corresponding cyclic product of projections is not asymptotically regular.
math.FA↗
Self-crossing geodesics
We discuss self-crossing patterns of closed geodesics on a convex surface.
math.DG↗
About every convex set in any generic Riemannian manifold
We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold (M,g). For example, if a convex set in (M,g) is bounded by a smooth hypersurface, then it is strictly convex.
math.DG↗
Moon in a puddle and the four-vertex theorem
We present a proof of the Moon in a puddle theorem, and use its key lemma to prove a generalization of the four-vertex theorem.
math.DG↗