Quantitative stratification of $F$-subharmonic functions

In this paper, we study the singular sets of $F$-subharmonic functions $u: B_{2}(0^{n})\rightarrow$\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}${\eta}^{n-p}(u)\cap B{1}(0^{n})))\leq C\eta^{-1}(\int_{B_{1+r}(0^{n})}\Delta u)r^{p}$. 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 ${\Theta(u,x)\geq c}$for$c>0$,where$\Theta(u,x)$isthedensityof$F$-subharmonicfunction$u$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}${\eta,r}^{k}(u))\cap B{1}(0^{n}))\leq Cr^{n-k-\eta}$.