Whitney's index formula in higher dimensions and Laplace integrals

维度$24$的椭圆亏格与string配边

Elliptic genus and string cobordism at dimension $24$

本文证明了维度24的string配边可由高阶指标不变量椭圆亏格确定，计算了其在椭圆亏格下的像，并由此得出特定曲率条件下紧致24维string流形必为边缘的结论。

端周期Dirac算子族的指标公式

An index formula for families of end-periodic Dirac operators

本文为端周期流形上的Dirac算子族建立了指标公式，通过转降公式和新的端周期eta形式，推广了已有的eta不变量。

Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure

Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\geq 3$, and let $φ_λ: Δ_M φ_λ+ λφ_λ= 0$ denote the Laplace eigenfunction on $\mathbb{M}$ corresponding to the eigenvalue $λ$. We show that $$H^{n-1}(\{ φ_λ=0\}) \leq C λ^α,$$ where $α>1/2$ is a constant, which depends on $n$ only, and $C>0$ depends on $\mathbb{M}$ . This result is a consequence of our study of zero sets of harmonic functions on $C^\infty$-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from …

具有Robin边界条件的平面区域的指数估计

Index estimates for planar domains with Robin boundary condition

本文研究在自然Robin边界条件下，平面区域上Laplace算子的指数，利用共形与谱技术及边界切向L^2调和向量场，得到指数下界与边界分支数的关系，为自由边界极小曲面中的指数估计提供内蕴对应。

Existence of harmonic maps and eigenvalue optimization in higher dimensions

We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold $(M^n,g)$ of dimension $n>2$ to any closed, non-aspherical manifold $N$ containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres $N=\mathbb{S}^k$, $k\geq 3$, we obtain a distinguished family of nonconstant harmonic maps $M\to \mathbb{S}^k$ of index at most $k+1$, with singular set of codimension at least …