arXiv research

The paper studies geometric structures of curvature radii on Riemannian manifolds.

In this paper, we obtain two-sided bounds for the volumes of the Aloff-Wallach spaces $W(p,q),$ compute maximal and minimal sectional curvature for the spaces $W(n,n+1),$ and use this information to estimate the injectivity radii: We derive an upper bound for the injectivity radii of $W(p,q)$ and a lower bound for the …

The study finds parametrizations for surfaces of revolution with a linear curvature ratio.

A Gauss equation is proved for subspaces of Alexandrov spaces of curvature bounded above by K. That is, a subspace of extrinsic curvature less than or equal to A, defined by a cubic inequality on the difference of arc and chord, has intrinsic curvature less than or equal to K+A^2. Sharp bounds on injectivity radii of s…

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of curvature is a second order system of partial differential equations which we sho…

The paper bounds radii and curvatures in Riemannian manifolds.

We prove that the normal curvatures of hyperspheres, the Rund curvature, and the Finsler curvature of circles in Hilbert geometry tend to 1 as the radii tend to infinity

We compute the series expansions for the normal curvatures of hyperspheres, the Finsler and Rund curvatures of circles in Funk geometry as the radii tend to infinity. These three curvatures are different at infinity in Funk geometry.

Researchers compute the full spectrum of Laplace operator on distance spheres in symmetric spaces.

We show that the 2-torus in ${\mathbb R}^3$ is a critical point of a sequence of functionals ${\cal F}_{n}$ ($n=1,2,3, \cdots$) defined over compact 2-surfaces in ${\mathbb R}^3$. When the Lagrange function ${\cal E}$ is a polynomial of degree $n$ of the mean curvature $H$ of the surface, the radii ($a,r$) of the 2-tor…

Minimal spectral radii found for specific matrix types.

The distance of an almost constant mean curvature boundary from a finite family of disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillation of the scalar mean curvature. This result allows one to quantitatively describe the geometry of volume-constrained stationary sets in capill…

We show that every effective action of a compact Lie group $K$ on a unit sphere $S^n$ admits an explicit orbit whose principal curvatures are bounded from above by $4\sqrt{14}$.

The article considers the problem of existence and uniqueness of centrally symmetrical convex body for which the projection curvature radius function coincides with a given flag function. A necessary and sufficient condition is found that ensures a positive answer. An algorithm for construction the body in question is …

In this article we pose the problem of existence and uniqueness of convex body for which the projection curvature radius function coincides with given function. We find a necessary and sufficient condition that ensures a positive answer to both questions and suggest an algorithm of construction of the body. Also we fin…

We obtain sharp lower bounds on the radii of inscribed balls for strictly convex isoperimetric domains lying in a 2-dimensional Alexandrov metric space of curvature bounded below. We also characterize the case when such bounds are attained.

Let T be the standard torus of revolution in R^3 with radii b and 1, 0<b<1. Let αbe a (p,q) torus curve on T. We show that there are points of zero curvature on αfor only one value of the variable radius of T, b=p^2/(p^2+q^2). The curve αhas non-vanishing curvature for all other values of b. Moreover, for this value of…

The paper studies a flow of convex hypersurfaces using anisotropic curvature functions.

For a convex domain $D$ bounded by the hypersurface $\partial D$ in a space of constant curvature we give sharp bounds on the width $R-r$ of a spherical shell with radii $R$ and $r$ that can enclose $\partial D$, provided that normal curvatures of $\partial D$ are pinched by two positive constants. Furthermore, in the …

Study on curvature in finitely generated groups, showing positive curvature in specific cases.

We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundar…

Paper improves neural network robustness certification with tighter radii estimates.

We study one extremal problem on the product of power of generalized inner radii of non-overlapping domains in $\mathbb{C}^{n}$.

Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in $H^{n+1}$, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with t…

Proves rigidity of geodesic balls in spheres under certain deformations.

New proof shows not all Salem numbers are growth rates of Coxeter groups.

We study the mean curvature flow of hypersurfaces in $\R^{n+1}$, with initial surfaces sufficiently close to the standard $n$-dimensional sphere. The closeness is in the Sobolev norm with the index greater than $\frac{n}{2}+1$ and therefore it does not impose restrictions of the mean curvature of the initial surface. W…

Study extends min-max eigenvalue results to $p$-energy and packing radii on Riemannian manifolds.

We construct a sequence of compact, oriented, embedded, two-dimensional surfaces of genus one into Euclidean 3-space with prescribed, almost constant, mean curvature of the form $H(X)=1+{A}{|X|^{-γ}}$ for $|X|$ large, when $A<0$ and $γ\in(0,2)$. Such surfaces are close to sections of unduloids with small necksize, fold…

Study nearest-neighbor radii under dependent sampling, finding they remain informative.

Generalizes a soul-bound for noncompact Alexandrov spaces.

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. …

The paper studies a flow of convex hypersurfaces expanding by their support and curvature functions.

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-…

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. …

Study properties of surfaces with nonvanishing third fundamental form.

Utilizing a splitting of geometric flows on surfaces introduced by Buzano and Rupflin, we present a general scheme to prove blow up criteria for such geometric flows. A vital ingredient is a new compactness theorem for families of metrics on surfaces with a uniform bound on their volumes, square integrals of their curv…

The paper explores harmonic and asymptotically harmonic Finsler manifolds and their properties.

The paper derives inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.

The paper studies geometric structures on tangent and sphere bundles over statistical manifolds.

Explicit BCH series radii found for special Banach-Malcev shift algebras.

New capillary Christoffel-Minkowski problem solved for half-space.

We consider complete Riemannian manifolds with a controlled growth of the covariant derivatives of Ricci curvatures up to order $k-2$ and a controlled decay of the injectivity radii. On such manifolds we construct distance-like functions with a control on covariant derivatives up to order $k$. Alternatively, the assump…

In the 1950's Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper we investigate conditions under which evolving a smooth rotationally symmetric sphere by a linear combination of its radii of …

Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.

The paper classifies and constructs rotational surfaces with constant astigmatism in space forms.

We give a new proof of the Gromov theorem: For any $C>0$ and integer $n>1$ there exists a function $Δ_{C,n}$ such that if the Gromov--Hausdorff distance between complete Riemannian $n$-manifolds $V$ and $W$ is not greater than $δ$, absolute values of their sectional curvatures $|K_σ|\leq C$, and their injectivity radii…

In the present paper, we study the infinitesimal symmetries of the model of two Riemannian manifolds $(M,g)$ and $(\hat M,\hat g)$ rolling without twisting or slipping. We show that, under certain genericity hypotheses, the natural bundle projection from the state space $Q$ of the rolling model onto $M$ is a principal …