We consider the problem of uniqueness of Ricci flows, starting from singular initial data. Even when the initial data is smooth and smoothly obtained, in the complete noncompact case uniqueness is not known in full generality. When starting from nonsmooth initial data, very little is known, even in the compact case. We obtain a uniqueness result for Ricci flows with scaling-invariant curvature bounds, a suitable noncollapsing assumption, and sectional curvature lower bounds, coming out of (possibly singular) Reifenberg spaces. As a consequence of our result, and existence results obtained by Simon--Topping and Lai, there is a canonical way to define a smooth structure on any Reifenberg uniformly volume-noncollapsed PIC2 limit space. Further, we obtain a diffeomorphic version of Perelman's stability theorem for these limit spaces.
In this paper, we prove a uniform pseudolocality theorem for smooth complete Ricci flows whose initial metrics have Ricci curvature bounded below and uniformly positive injectivity radius. No a priori curvature bound is assumed on the flow. As a consequence, we obtain strong uniqueness for smooth complete Ricci flows f...
In this paper, we study strong uniqueness for smooth instantaneously complete Chern-Ricci flows on noncompact complex manifolds. Under suitable reference metric and positivity assumptions, we establish strong uniqueness for initial Hermitian metrics which are K\"ahler outside a compact set. The initial metric may be in...
We prove Petrunin's smoothing conjecture in all dimensions: every compact Euclidean polyhedral space without boundary and with nonnegative Alexandrov curvature is a Gromov-Hausdorff limit of smooth Riemannian orbifolds with geometrically nonnegative curvature. More strongly, the approximating metrics are positive-time...
We consider singular Liouville systems on closed Riemann surfaces with a common sign-changing coefficient. Direct continuation from a positive coefficient is obstructed by the unavoidable appearance of degenerate zeroes along such a homotopy. We instead identify an intrinsic ray in the parameter space on which the syst...
Mean curvature flow (MCF) is an effective tool for investigating the geometry and topology of submanifolds under suitable curvature pinching conditions. While most existing convergence results in complex projective spaces rely on pointwise curvature assumptions, much less is known about the case of integral curvature c...
We prove that when the J-equation does not admit a smooth solution, there is always only a finite number of obstructing optimally destabilizing curves and divisors on compact K\"ahler manifolds of any dimension. For threefolds, this establishes the unconditional finiteness of all optimally destabilizing subvarieties. W...
P. Sivaram, Zakarias Sjostrom Dyrefelt· 4 citations· ⚡1
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