Absence of bubbling phenomena for non-convex anisotropic nearly umbilical and quasi-Einstein hypersurfaces
Abstract We prove that, for every closed (not necessarily convex) hypersurface Σ in ℝ n + 1 {\mathbb{R}^{n+1}} and every p > n {p>n} , the L p {L^{p}} -norm of the trace-free part of the anisotropic second fundamental form controls from above the W 2 , p {W^{2,p}} -closeness of Σ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime p ≤ n {p\leq n} , the lack of convexity assumptions may lead in general to bubbling phenomena.Moreover, we obtain a stability theorem for quasi-Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting.
Authors:
;
Award ID(s):
Publication Date:
NSF-PAR ID:
10326411
Journal Name:
Journal für die reine und angewandte Mathematik (Crelles Journal)
Volume:
2021
Issue:
780
Page Range or eLocation-ID:
1 to 40
ISSN:
0075-4102
4. Abstract Recently, Dvořák, Norin, and Postle introduced flexibility as an extension of list coloring on graphs (J Graph Theory 92(3):191–206, 2019, https://doi.org/10.1002/jgt.22447 ). In this new setting, each vertex v in some subset of V ( G ) has a request for a certain color r ( v ) in its list of colors L ( v ). The goal is to find an L coloring satisfying many, but not necessarily all, of the requests. The main studied question is whether there exists a universal constant $$\varepsilon >0$$ ε > 0 such that any graph G in some graph class $$\mathscr {C}$$ C satisfies at least $$\varepsilon$$ ε proportion of the requests. More formally, for $$k > 0$$ k > 0 the goal is to prove that for any graph $$G \in \mathscr {C}$$ G ∈ C on vertex set V , with any list assignment L of size k for each vertex, and for every $$R \subseteq V$$ R ⊆ V and a request vector $$(r(v): v\in R, ~r(v) \in L(v))$$ ( r ( v ) : v ∈ R , r ( v ) ∈ L ( v ) ) , there exists an L -coloring of Gmore »
The free multiplicative Brownian motion$$b_{t}$$${b}_{t}$is the large-Nlimit of the Brownian motion on$$\mathsf {GL}(N;\mathbb {C}),$$$\mathrm{GL}\left(N;C\right),$in the sense of$$*$$$\ast$-distributions. The natural candidate for the large-Nlimit of the empirical distribution of eigenvalues is thus the Brown measure of$$b_{t}$$${b}_{t}$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$\Sigma _{t}$$${\Sigma }_{t}$that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$W_{t}$$${W}_{t}$on$$\overline{\Sigma }_{t},$$${\overline{\Sigma }}_{t},$which is strictly positive and real analytic on$$\Sigma _{t}$$${\Sigma }_{t}$. This density has a simple form in polar coordinates:\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$\begin{array}{c}{W}_{t}\left(r,\theta \right)=\frac{1}{{r}^{2}}{w}_{t}\left(\theta \right),\end{array}$where$$w_{t}$$${w}_{t}$is an analytic function determined by the geometry of the region$$\Sigma _{t}$$${\Sigma }_{t}$. We show also that the spectral measure of free unitary Brownian motion$$u_{t}$$${u}_{t}$is a “shadow” of the Brown measure of$$b_{t}$$${b}_{t}$, precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.