For every $n\ge3$, we show, by a family of explicit entire real-analytic solutions, that, for every subcritical phase and every $0<\alpha\le1$, oscillation-based interior $C^\alpha$ estimates fail for the special Lagrangian equation in $\mathbb{R}^n$. In fact, this family has no common interior modulus of continuity.
Let $(M^n,g)$ be a complete connected noncompact smooth Riemannian manifold without boundary, with $n\ge3$ and positive scalar curvature. We prove that $M$ admits a complete smooth metric of uniformly positive scalar curvature if $$ \liminf_{d_g(o,x)\to\infty}d_g(o,x)^2R_g(x)>\frac{n-1}{n} $$ for a fixed point $o\in M$...
Let $B=B(p, 1)\subset\mathbb{R}^n$ be a unit ball and $n\ge 3$. We prove that there are positive constants $c$ and $C$, depending only on $n$, such that every non-zero real-valued harmonic function $u: 4B\to \mathbb{R}$ with $u(p)=0$ satisfies $$ \mathcal{H}^{n-1}\bigl(\{u=0\}\cap 2B\bigr)\ge C\mathcal{N}, $$ and $$ \m...
Let $4\le n\le7$ and let $(M^n,g)$ be a complete, connected, orientable, noncompact Riemannian manifold of positive scalar curvature. We prove that if the asymptotic quadratic scalar curvature coefficient of $g$ is greater than $(n-1)/n$, then $M$ carries a complete smooth metric whose scalar curvature is at least one....
Let $u$ be a non-zero real-valued harmonic function in $B_4\subset\mathbb{R}^n$ with $n\geq3$ and $u(0)=0$. We prove that $$\mathcal{H}^{n-1}\bigl(\{u(x)=0\}\cap B_2\bigr)\ge C_n\mathcal{N},$$ where $C_n$ is a positive constant depending only on $n$, and $\mathcal{N}$ is the doubling index defined by $$\mathcal{N}=\log...
Let $(M^4,g)$ be a complete four-dimensional Riemannian manifold. First, if the $Q$-curvature $Q_g\geq 6k^2$ and scalar curvature $R_g\geq -12k$ for some positive constant $k$, then $(M^4,g)$ is either Einstein with $Ric_g=-3kg$ or compact with $R_g\ge 12k$. As a corollary, the fundamental group $\pi_1(M^4)$ satisfies...
Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the vo...
