In this paper, we derive some local a priori estimates for the Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow on R3, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t) ≡ E.

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This paper studies affine Deligne-Lusztig varieties $X_{w}(b)$ in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of $X_{w}(b)$ for a minimal length element $w$ in the conjugacy class of an extended affine Weyl group, generalizing one of the main results in \cite{HL} to the affine case. We then provide a reduction method that relates the structure of $X_{w}(b)$ for arbitrary elements $w$ in the extended affine Weyl group to those associated with minimal length elements. Based on this reduction, we establish a connection between the dimension of affine Deligne-Lusztig varieties and the degree of the class polynomial of affine Hecke algebras. As a consequence, we prove a conjecture of G\"ortz, Haines, Kottwitz and Reuman in \cite{GHKR}.

Consider $N$ bosons in a finite box  $ \Lambda = [0,L]^3 \subset R^3$ interacting via a two-body smooth repulsive short range potential. We construct a variational state which gives the following upper bound on the ground state energy per particle $$ \bar{lim|_{\rho \to 0} \bar{lim|_{L \to \infty, N/L^3 \to \rho} ( \ frac{e_0 (\rho) - 4\pi a \rho }{ (4\pi a)^{5/2} (\rho)^{3/2} } ) \le \frac{16}{15 \pi ^2} , $$ where $a$ is the scattering length of the potential. Previously, an upper bound of the form $C16/15\pi^2$ for some constant $C >1$ was obtained in (Erdös et al. in Phys. Rev. A 78:053627, 2008). Our result proves the upper bound of the prediction by Lee and Yang (Phys. Rev. 105(3):1119–1120, 1957) and Lee et al. (Phys. Rev. 106(6):1135–1145, 1957).

Necessary and sufficient conditions are given for the existence of solutions to the discrete Lp Minkowski problem for the critical case where 0 < p < 1.