We prove a sharp volume gap estimate: if an $n$-dimensional compact K\"ahler manifold $(X, \omega)$ satisfies $\mathrm{Ric}(\omega)\ge (n+1)\omega$ and $X\not\cong \mathbb{P}^n$, then $\mathrm{vol}(X, \omega)\le \frac{2n^n}{(n+1)^n}\mathrm{vol}(\mathbb{P}^n,\omega_{\mathrm{FS}})=\frac{2^{n+1} \, \pi^n \, n^n}{(n+1)^n}$. Moreover $\mathrm{vol}(X, \omega)= \frac{2^{n+1} \, \pi^n \, n^n}{(n+1)^n}$ occurs if and only if $(X, \omega)$ is biholomorphically isometric to the K\"ahler-Einstein metric on the quadric hypersurface $Q^n$ or on the product $\mathbb{P}^1\times \mathbb{P}^{n-1}$. We also obtain sharp volume gap estimates for K-semistable toric log Fano pairs.
In this paper, we study the sharp Sobolev-type inequality associated with the Born--Infeld energy on the lattice graph \(\mathbb Z^N\), \(N\geq3\): \[ \frac12\sum_{x\in\mathbb Z^N}\sum_{y\sim x} \left(1-\sqrt{1-|\nabla_{xy}u|^2}\right) \geq C_{N,\alpha} \left(\sum_{x\in\mathbb Z^N}|u(x)|^\alpha\right)^{\frac{N}{N+\alph...
Let $\mathcal{X}$ be a complex manifold and $X$ a compact complex manifold with boundary in $\mathcal{X}$. For a complex Lie group $G$ and a regularity class $$\mathfrak{r}\in \big\{\mathcal{C}^k|\ k\in\mathbb{N}\cup\{\infty\}\big\}\cup\big\{\Lambda^r_{\rm loc}|\ r\in (0,\infty)\big\} $$ we define the sheaf of groups $...
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...
For any complete noncompact manifolds $(M^n, g)$ of nonnegative sectional curvature, with $3\le n\le6$, we obtain $\displaystyle \lim_{r\to\infty}r^{2-n}\int_{B_p(r)}\Sc_g\,dV_g\le8\pi\omega_{n-2}(1-\AVR(M^n,g))$ and the corresponding rigidity. The latter bound and its equality classification hold in $(M^n, g)$ with po...
We prove for the Euclidean space $\left(\mathbb{R}^n, \|.\|_2\right)$ the existence of subsets $A^n \subseteq \mathbb{R}^n$ with $d_H(\mathbb{R}^n, A^n)$ arbitrarily small, such that $d_{GH}(\mathbb{R}^n, A^n) = \sqrt{\frac{n+1}{2n}}d_{H}(\mathbb{R}^n, A^n)$, showing that the lower bound $d_{GH}(\mathbb{R}^n, A^n) \geq...
Let \[ \xi\!\left(\frac12+z\right) =\sum_{n\geq 0}\frac{\gamma(n)}{n!}z^{2n}, \qquad J^{d,n}(X) =\sum_{j=0}^{d}\binom dj\gamma(n+j)X^j . \] The Riemann hypothesis is equivalent to the hyperbolicity of $J^{d,n}$ for every $d,n\geq0$. We prove that there is an absolute constant $K>0$ such that \[ n^3\log^2(n+2)\geq Kd^5...
J. Holland· 0 citations
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