Global transversal stability of Euclidean planes under skew mean curvature flow evolutions

Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds

We prove that a continuous potential $q$ can be constructively determined from the knowledge of the Dirichlet-to-Neumann map for the perturbed biharmonic operator $Δ_g^2+q$ on a conformally transversally anisotropic Riemannian manifold of dimension $\ge 3$ with boundary, assuming that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [51]. In particular, our result is applicable and new in the case of smooth bounded domains in the $3$-dimensional Euclidean space as well as in the case of $3$-dimen…

各向异性Calderón问题：欧几里得度量附近的刚性

The anisotropic Calderón problem: rigidity near the Euclidean metric

该文证明三维及以上区域中各向异性Calderón问题在欧几里得度量附近的刚性：若光滑度量充分接近欧几里得度量且具有相同Dirichlet-to-Neumann映射，则必相差一个固定边界的微分同胚，方法结合调和坐标与Carleman估计。

Skew Mean Curvature Flow

The skew mean curvature flow(SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space $\mathbb{R}^4$. A Sobolev-type embedding theorem for the second fundamental forms of two dimensional surfaces is also proved, which might be of independent interest.