arXiv research

Extends symmetry and rigidity to surfaces with soap film-like singularities.

The paper studies how points and lines can move while preserving incidences.

New rigidity results for manifolds with maximal symmetry rank and positive intermediate Ricci curvature.

Proves rigidity of maps between balls with Hölder boundary continuity.

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.

New rigidity found for 3D warped product domains.

The paper shows symmetries of a geometric space for Coxeter groups.

Study on cylindrical handlebody-knots with symmetry and rigidity properties.

The goal of this paper is twofold. First we prove a rigidity estimate, which generalises the theorem on geometric rigidity of Friesecke, James and Müller to 1-forms with non-vanishing exterior derivative. Second we use this estimate to prove a kind of spontaneous breaking of rotational symmetry for some models of cryst…

Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.

We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of the complete graph K_{4r+3} in S^3.

Theory of symmetric rigidity in hyperbolic geometry.

Study on scalar-flat Kahler 4-manifolds with a continuous symmetry.

Study scattering rigidity on stationary manifolds using geodesics.

Discussing rigidity properties of conics, inspired by billiards in ellipses.

We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear representation results that may be interesting in their own right. The same techniques immedi…

In this paper we study a novel class of parabolic geometries which we call parabolic geometries of Monge type. These parabolic geometries are defined by special gradings of simple Lie algebras, namely, gradings with the property that their -1 component contains a nonzero co-dimension 1 abelian subspace whose bracket wi…

Study on symmetries and equilibria in Poisson manifolds, with applications to rigid body dynamics.

Paper solves Minkowski problem for anisotropic p-torsional rigidity.

Study rigidifies Einstein manifolds with symmetry, proving conjecture.

New model simplifies symmetry handling in generative AI.

A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwi…

Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the $\nabla u=f(u)$ Poisson's equation, which has a subalgebra isomorphic to the $3-$dimensional special Euclidean group ${\rm SE}(3)$ or group of rigid motions of ${\Bbb R}^3$. Looking the adjoint representation of ${\rm SE}(3)…

Proves symmetries of extremal horizons in spacetimes.

Survey on symmetry in manifold structures.

Study $SO(3)$-invariant $G_2$-cobordisms on 6-manifolds.

Paper finds isometric timelike minimal surfaces with unique properties.

We study gradient Ricci solitons with maximal symmetry. First we show that there are no non-trivial homogeneous gradient Ricci solitons. Thus the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the…

Paper proves rigidity of CMC surfaces in curved 3-manifolds.

In this paper, we give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain four dimensional fiber bundles by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau. As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree o…

Paper compares Lagrangian reduction methods for rigid body systems.

In this paper, we prove that the even solution of the mean field equation $Δu=λ(1-e^u)$ on $S^2$ must be axially symmetric when $4<λ\leq 8$. In particular, zero is the only even solution for $λ=6$. This implies the rigidity of Hawking mass for stable constant mean curvature(CMC) sphere with even symmetry.

We prove a lower bound for the first eigenvalue of the sub-Laplacian on sub-Riemannian manifolds with transverse symmetries. When the manifold is of H-type, we obtain a corresponding rigidity result: If the optimal lower bound for the first eigenvalue is reached, then the manifold is equivalent to a 1 or a 3-Sasakian s…

Unique steady and expanding solitons with spherical links identified.

We derive modified Perelman-type monotonicity formulas for solutions to the generalized Ricci flow equation with symmetry on principal bundles, which lead to rigidity and classification results for nonsingular solutions.

Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. W…

Study rigidity in Riemannian manifolds using Pohozoaev and P-function approaches.

Study proves symmetry of bounded domains in Riemannian manifolds.

We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyper…

We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some r…

This thesis is divided into two parts. In the first part we study completely integrable systems, and their underlying structures, in detail. We study their deformation theory and the different equivalence relations surrounding it. We motivate the definition of weak equivalence (found in the literature) by studying diff…

In this note, we study symmetry of solutions of the elliptic equation \begin{equation*} -Δ_{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2}, \end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only const…

We consider a generalization of the notion of a natural mechanical system to the case of additional forces of gyroscopic type. Such forces appear, for example, as a result of global reduction of a natural system with symmetry. We study symmetries in the systems with gyroscopic forces to find out when these systems admi…

Rigidity theorem for critical points of Allen-Cahn equation on S³.

The paper explores rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.

The paper proves rigidity results for manifolds with special holonomy.

Moitvated in part by [3], in this note we obtain a rigidity result for globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a timelike conformal Killing vector field. Specifically, we show that if M is a Ricci flat, timelike geodesically complete spacetime with compact Cauchy surfaces that admits a t…

Study proves rigid spectral properties of planets with metric discontinuities.