IRIS Uni Torinohttps://iris.unito.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Wed, 26 Jan 2022 04:17:55 GMT2022-01-26T04:17:55Z10171On Type-I singularities in Ricci flowhttp://hdl.handle.net/2318/1701039Titolo: On Type-I singularities in Ricci flow
Abstract: We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber. As a by-product we conclude that the volume of a finite-volume singular set vanishes at the singular time. We also define a notion of density for Type I Ricci flows and use it to prove a regularity theorem reminiscent of White's partial regularity result for mean curvature flow.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2318/17010392011-01-01T00:00:00ZRicci flow coupled with harmonic map flowhttp://hdl.handle.net/2318/1701037Titolo: Ricci flow coupled with harmonic map flow
Abstract: We investigate a new geometric flow which consists of a coupled system of the Ricci flow on a closed manifold M with the harmonic map flow of a map phi from M to some closed target manifold N with a (possibly time-dependent) positive coupling constant alpha. This system can be interpreted as the gradient flow of an energy functional F_alpha which is a modification of Perelman's energy F for the Ricci flow, including the Dirichlet energy for the map phi. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of phi a-priori - without any assumptions on the curvature of the target manifold N - by choosing alpha large enough. Moreover, if alpha is bounded away from zero it suffices to bound the curvature of (M,g(t)) to also obtain control of phi and all its derivatives - a result which is clearly not true for alpha = 0. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of an entropy functional W_alpha similar to Perelman's Ricci flow entropy W and of so-called reduced volume functionals. We then apply these monotonicity results to rule out non-trivial breathers and geometric collapsing at finite times.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/2318/17010372012-01-01T00:00:00ZDynamical stability and instability of Ricci-flat metricshttp://hdl.handle.net/2318/1701045Titolo: Dynamical stability and instability of Ricci-flat metrics
Abstract: In this short article, we improve the dynamical stability and instability results for Ricci-flat metrics under Ricci flow proved by Sesum and Haslhofer, getting rid of the integrability assumption.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/2318/17010452014-01-01T00:00:00ZBubbling analysis and geometric convergence results for free boundary minimal surfaceshttp://hdl.handle.net/2318/1721873Titolo: Bubbling analysis and geometric convergence results for free boundary minimal surfaces
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/2318/17218732019-01-01T00:00:00ZThe Chern-Gauss-Bonnet formula for singular non-compact four-dimensional manifoldshttp://hdl.handle.net/2318/1726079Titolo: The Chern-Gauss-Bonnet formula for singular non-compact four-dimensional manifolds
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/2318/17260792019-01-01T00:00:00ZQualitative and quantitative estimates for minimal hypersurfaces with bounded index and areahttp://hdl.handle.net/2318/1701049Titolo: Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area
Abstract: We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index, the total curvature along the sequence is quantised in terms of the total curvature of some limit surface, plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in Euclidean space - all of which are finite. Thus, we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/2318/17010492018-01-01T00:00:00ZDifferential Harnack Inequalities and the Ricci Flowhttp://hdl.handle.net/2318/1701023Titolo: Differential Harnack Inequalities and the Ricci Flow
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/2318/17010232006-01-01T00:00:00ZMonotone volume formulas for geometric flowshttp://hdl.handle.net/2318/1701021Titolo: Monotone volume formulas for geometric flows
Abstract: We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter being non-decreasing along the flow. In the case where S=Ric is the Ricci curvature of M, the result corresponds to Perelman's well-known reduced volume monotonicity for the Ricci flow. Some other examples are given in the second section of this article, the main examples and motivation for this work being List's extended Ricci flow system, the Ricci flow coupled with harmonic map heat flow and the mean curvature flow in Lorentzian manifolds with nonnegative sectional curvatures. With our approach, we find new monotonicity formulas for these flows.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/2318/17010212010-01-01T00:00:00ZA note on the compactness theorem for 4d Ricci Shrinkershttp://hdl.handle.net/2318/1701047Titolo: A note on the compactness theorem for 4d Ricci Shrinkers
Abstract: In arXiv:1005.3255 we proved an orbifold Cheeger-Gromov compactness theorem for complete 4d Ricci shrinkers with a lower bound for the entropy, an upper bound for the Euler characterisic, and a lower bound for the gradient of the potential at large distances. In this note, we show that the last two assumptions in fact can be removed. The key ingredient is a recent estimate of Cheeger-Naber arXiv:1406.6534.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/2318/17010472015-01-01T00:00:00ZA Compactness Theorem for Complete Ricci Shrinkershttp://hdl.handle.net/2318/1701042Titolo: A Compactness Theorem for Complete Ricci Shrinkers
Abstract: We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss-Bonnet with cutoff argument.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2318/17010422011-01-01T00:00:00Z