Quantitative stratification of $F$-subharmonic functions

Abstract

In this paper, we study the singular sets of $F$-subharmonic functions $u:B_2(0^n)\to \mathbf{R}$, where $F$ is a subequation. The singular set $\mathcal{S}(u)\subset B_2(0^n)$ has a stratification ${\mathcal{S}}^0(u)\subset {\mathcal{S}}^1(u)\subset \cdots \subset {\mathcal{S}}^k(u)\subset \cdots \subset \mathcal{S}(u)$, where $x\in {\mathcal{S}}^k(u)$ if no tangent function to $u$ at $x$ is $(k+1)$-homogeneous. We define the quantitative stratification ${\mathcal{S}}_{\eta ,r}^k(u)$ and ${\mathcal{S}}_{\eta }^k(u)={\cap }_r{\mathcal{S}}_{\eta ,r}^k(u)$. When homogeneity of tangents holds for $F$, we prove that $di{m}_H{\mathcal{S}}^k(u)\le k$ and $\mathcal{S}(u)={\mathcal{S}}^{n-p}(u)$, where $p$ is the Riesz characteristic of $F$. And for the top quantitative stratification ${\mathcal{S}}_{\eta }^{n-p}(u)$, we have the Minkowski estimate $\text{Vol}(B_r({\mathcal{S}}_{\eta }^{n-p}(u)\cap B_1(0^n)))\le C{\eta }^{-1}({\int }_{B_{1+r}(0^n)}\Delta u)r^p$. When uniqueness of tangents holds for $F$, we show that $S_{\eta }^k(u)$ is $k$-rectifiable, which implies ${\mathcal{S}}^k(u)$ is $k$-rectifiable. When strong uniqueness of tangents holds for $F$, we introduce the monotonicity condition and the notion of $F$-energy. By using refined covering argument, we obtain a definite upper bound on the number of $\{\Theta (u,x)\ge c\}$ for $c>0$, where $\Theta (u,x)$ is the density of $F$-subharmonic function $u$ at $x$. Geometrically determined subequations $F(\mathbb{G})$ is a very important kind of subequation (when $p=2$, homogeneity of tangents holds for $F(\mathbb{G})$; when $p>2$, uniqueness of tangents holds for $F(\mathbb{G})$). By introducing the notion of $\mathbb{G}$-energy and using quantitative differentation argument, we obtain the Minkowski estimate of quantitative stratification $\text{Vol}(B_r({\mathcal{S}}_{\eta ,r}^k(u))\cap B_1(0^n))\le C{r}^{n-k-\eta }$.

38 pages; added more details; Lemma 7.11 and Section 8.1 are rewritten; corrected typos; references updated; other places have also some small changes

References in corpus (1)

• Energy Identity for Stationary Yang Mills