Related papers: Llarull type theorems for bands in Three and Four …
In this paper we present a self-contained combinatorial proof of the lower bound theorem for normal pseudomanifolds, including a treatment of the cases of equality in this theorem. We also discuss McMullen and Walkup's generalised lower…
We produce new examples of Riemannian manifolds with scalar curvature lower bounds and collapsing behavior along codimension 2 submanifolds. Applications of this construction are given, primarily on questions concerning the stability of…
We establish Bernstein-type theorems for entire constant mean curvature graphs in the three-dimensional light cone $\mathbb{Q}^3_+$ over the horosphere under the assumption that the Gaussian curvature $K$ is bounded below, by showing that…
The Sperner and Tucker lemmas are combinatorial analogous of the Brouwer and Borsuk - Ulam theorems with many useful applications. These classic lemmas are concerning labellings of triangulated discs and spheres. In this paper we show that…
In this paper, we proved the Normal Scalar Curvature Conjecture and the Bottcher-Wenzel Conjecture. We also established some new pinching theorems for minimal submanifolds in spheres.
Llarull's scalar curvature rigidity theorem states that a 1-Lipschitz map $f: M\to S^n$ from a closed connected Riemannian spin manifold $M$ with scalar curvature $\mathrm{scal}\ge n(n-1)$ to the standard sphere $S^n$ is an isometry if the…
Recently, Zhang \cite{Zh20} and Li-Su-Wang-Zhang \cite{LSWZ24+} generalized Llarull's theorem to the noncompact complete spin manifold. In this paper, we further extend their results to the noncompact manifold with compact boundary.
A framework allowing for perturbative calculations to be carried out for quantum field theories with arbitrary smoothly curved boundaries is described. It is based on an expansion of the heat kernel derived earlier for arbitrary mixed…
An old theorem, due to Graustein, asserts that the average curvature of a plane oval is attained at least at four points. We present a proof by way of wave propagation and extend this result to the spherical and hyperbolic geometries - in…
In this work we construct a sequence of Riemannian metrics on the three-sphere with scalar curvature greater than or equal to $6$ and arbitrarily large widths. Our procedure is based on the connected sum construction of positive scalar…
We consider the case with boundary of the classical Kazdan-Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular…
Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. For smooth curves and polygonal lines in the plane, a formula relating the number of…
The classic Ky Fan theorem is a combinatorial equivalent of Borsuk-Ulam theorem. It is a generalization and extension of Tucker's lemma and, just like its predecessor, it pinpoints important properties of antipodal colorings of vertices of…
We study Borsuk-Ulam type results for the loopspace of an euclidean sphere without loops equal to their inverses.
The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras) are introduced. Cohomological range leads to the concepts of derived bounded algebras and…
We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors, and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to…
The virial theorem is related to the dilatation properties of bound states. This is realized, in particular, by the Landau-Lifshitz formulation of the relativistic virial theorem, in terms of the trace of the energy-momentum tensor. We…
We consider the mean curvature rigidity problem of an equatorial zone on a sphere which is symmetric about the equator with width $2w$. There are two different notions on rigidity, i.e. strong rigidity and local rigidity. We prove that for…
Using Schauder's theory for linear elliptic partial differential equations in two independent variables and fundamental estimates for univalent mappings due to E. Heinz we establish an upper bound of the Gaussian curvature of…
By using Gromov's $\mu$-bubble technique, we show that the $3$-dimensional spherical caps are rigid under perturbations that do not reduce the metric, the scalar curvature, and the mean curvature along its boundary. Several generalizations…