Rigidity estimate for two incompatible wells

Two Rigidity Theorems for Gravity

In this paper we prove two theorems on initial data for general relativity. The results present “rigidity phenomena” for the extrinsic curvature, caused by the negativity of the scalar curvature. More precisely, Theorem 1 states that in the case of non-vacuum initial data if the metric has everywhere non-positive scalar curvature then the extrinsic curvature cannot be compactly supported. Theor...

Maxwell-independence: a new rank estimate for 3D rigidity matroids

Maxwell’s condition states that the edges of a graph are independent in its d-dimensional generic rigidity matroid only if (a) the number of edges does not exceed d|V | − ( d+1 2 ) , and (b) this holds for every induced subgraph. We call such graphs Maxwell-independent in d dimensions. Laman’s theorem shows that the converse holds in 2D. While the converse is false in 3D, we answer the followin...

Geometric Rigidity for Incompatible Fields and an Application to Strain-gradient Plasticity

In this paper we show that a strain-gradient plasticity model arises as the Γ-limit of a nonlinear semi-discrete dislocation energy. We restrict our analysis to the case of plane elasticity, so that edge dislocations can be modelled as point singularities of the strain field. A key ingredient in the derivation is the extension of the rigidity estimate [10, Theorem 3.1] to the case of fields β :...

für Mathematik in den Naturwissenschaften Leipzig Rigidity Estimate

Isometric rigidity in codimension two

In the local theory of submanifolds a fundamental but difficult problem is to describe the isometrically deformable isometric immersions f : M → R into Euclidean space with low codimension p if compared to the dimension n ≥ 3 of the Riemannian manifold. Moreover, one would like to understand the set of all possible isometric deformations. Submanifolds in low codimension are generically rigid si...