Title: Volume Rigidity of Simplicial Manifolds Show Chinese title

Title: 单纯流形的体积刚性

Abstract: Classical results of Cauchy and Dehn imply that the 1-skeleton of a convex polyhedron $P$ is rigid i.e. every continuous motion of the vertices of $P$ in $\mathbb R^3$ which preserves its edge lengths results in a polyhedron which is congruent to $P$. This result was extended to convex poytopes in $\mathbb R^d$ for all $d\geq 3$ by Whiteley, and to generic realisations of 1-skeletons of simplicial $(d-1)$-manifolds in $\mathbb R^{d}$ by Kalai for $d\geq 4$ and Fogelsanger for $d\geq 3$. We will generalise Kalai's result by showing that, for all $d\geq 4$ and any fixed $1\leq k\leq d-3$, every generic realisation of the $k$-skeleton of a simplicial $(d-1)$-manifold in $\mathbb R^{d}$ is volume rigid, i.e. every continuous motion of its vertices in $\mathbb R^d$ which preserves the volumes of its $k$-faces results in a congruent realisation. In addition, we conjecture that our result remains true for $k=d-2$ and verify this conjecture when $d=4,5,6$. Show Chinese abstract

Abstract: 柯西和德恩的经典结果表明，凸多面体$P$的 1-骨架是刚性的，即。 $P$的顶点在$\mathbb R^3$中的每一个保持其边长的连续运动都会导致一个与$P$全等的多面体。 此结果被Whiteley扩展到$\mathbb R^d$中的所有$d\geq 3$凸多面体，以及Kalai在$\mathbb R^{d}$中对单纯$(d-1)$-流形的1-骨架的通用实现，对于$d\geq 4$和Fogelsanger对于$d\geq 3$。 我们将通过证明，对于所有$d\geq 4$和任何固定的$1\leq k\leq d-3$，在$\mathbb R^{d}$中，单纯$(d-1)$-流形的$k$-骨架的每个一般性实现都是体积刚性的，即： 在其顶点在$\mathbb R^d$中的连续运动中，若保持其$k$-面的体积不变，则会导致全等的实现。 此外，我们猜想我们的结果对于$k=d-2$仍然成立，并在$d=4,5,6$时验证了这个猜想。