Showing papers on "Ricci flow published in 2023"
TL;DR: In this article , a compactness theory for super Ricci flows was developed, which lays the foundations for the partial regularity theory in Bamler's this article paper, which implies that any sequence of super Riccis flows of the same dimension that is pointed in an appropriate sense subsequentially converges to a certain type of synthetic flow, called a metric flow.
Abstract: We develop a compactness theory for super Ricci flows, which lays the foundations for the partial regularity theory in Bamler (Structure Theory of Non-collapsed Limits of Ricci Flows, arXiv:2009.03243 , 2020). Our results imply that any sequence of super Ricci flows of the same dimension that is pointed in an appropriate sense subsequentially converges to a certain type of synthetic flow, called a metric flow. We will study the geometric and analytic properties of this limiting flow, as well as the convergence in detail. We will also see that, under appropriate local curvature bounds, a limit of Ricci flows can be decomposed into a regular and singular part. The regular part can be endowed with a canonical structure of a Ricci flow spacetime and we have smooth convergence on a certain subset of the regular part.
3 citations
TL;DR: In this article , the authors generalize the Bochner formula for the heat flow on evolving manifolds to an infinite-dimensional version for martingales on parabolic path space.
Abstract: We generalize the classical Bochner formula for the heat flow on evolving manifolds $$(M,g_{t})_{t \in [0,T]}$$ to an infinite-dimensional Bochner formula for martingales on parabolic path space $$P{\mathcal {M}}$$ of space-time $${\mathcal {M}} = M \times [0,T]$$ . Our new Bochner formula and the inequalities that follow from it are strong enough to characterize solutions of the Ricci flow. Specifically, we obtain characterizations of the Ricci flow in terms of Bochner inequalities on parabolic path space. We also obtain gradient and Hessian estimates for martingales on parabolic path space, as well as condensed proofs of the prior characterizations of the Ricci flow from Haslhofer–Naber (J Eur Math Soc 20(5):1269–1302, 2018a). Our results are parabolic counterparts of the recent results in the elliptic setting from Haslhofer–Naber (Commun Pure Appl Math 71(6):1074–1108, 2018b).
2 citations
TL;DR: In this article , the uniqueness of the Ricci flow in higher dimensions was studied and it was shown that the only noncompact ancient solution up to isometry are a family of shrinking cylinders, a quotient thereof, or the Bryant soliton.
Abstract: We extend the second part of \cite{Bre20} on the uniqueness of ancient $\kappa$-solutions to higher dimensions. In dimensions $n \geq 4$, an ancient $\kappa$-solution is a nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2; has bounded curvature; and is $\kappa$-noncollapsed. We show that the only noncompact ancient $\kappa$-solutions up to isometry are a family of shrinking cylinders, a quotient thereof, or the Bryant soliton.
2 citations
1 citations
TL;DR: In this paper , an attempt to conciliate the Ricci flow conjecture and the Cobordism conjecture, stated as refinements of the Swampland distance conjecture and of the No global symmetries conjecture respectively, is presented.
Abstract: A bstract In the following work, an attempt to conciliate the Ricci flow conjecture and the Cobordism conjecture, stated as refinements of the Swampland distance conjecture and of the No global symmetries conjecture respectively, is presented. This is done by starting from a suitable manifold with trivial cobordism class, applying surgery techniques to Ricci flow singularities and trivialising the cobordism class of one of the resulting connected components via the introduction of appropriate defects. The specific example of $$ {\varOmega}_4^{SO} $$ Ω 4 SO is studied in detail. A connection between the process of blowing up a point of a manifold and that of taking the connected sum of such with ℂℙ n is explored. Hence, the problem of studying the Ricci flow of a K 3 whose cobordism class is trivialised by the addition of 16 copies of ℂℙ 2 is tackled by applying both the techniques developed in the previous sections and the classification of singularities in terms of ADE groups.
1 citations
TL;DR: In this paper , a generalized soliton on a Riemannian manifold was studied and sufficient conditions under which it reduces to a quasi-Einstein manifold were derived for the Euclidean space.
Abstract: In this paper, we initiate the study of a generalized soliton on a Riemannian manifold, we find a characterization for the Euclidean space, and in the compact case, we find a sufficient condition under which it reduces to a quasi-Einstein manifold. We also find sufficient conditions under which a compact generalized soliton reduces to an Einstein manifold. Note that Ricci solitons being self-similar solutions of the heat flow, this topic is related to the symmetry in the geometry of Riemannian manifolds. Moreover, generalized solitons being generalizations of Ricci solitons are naturally related to symmetry.
1 citations
TL;DR: In this paper , the authors presented new gradient estimates for positive solutions to a class of nonlinear elliptic equations involving the f-Laplacian on a smooth metric measure space.
Abstract: In this article, we present new gradient estimates for positive solutions to a class of non‐linear elliptic equations involving the f‐Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet–Zhang and Hamilton types, respectively, and are established under natural lower bounds on the generalised Bakry–Émery Ricci curvature tensor. From these estimates, we derive amongst other things Harnack inequalities and general global constancy and Liouville‐type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed.
1 citations
30 Jun 2023
TL;DR: The mathematics required to analyse higher dimensional curved spaces and space-times is developed in this paper , where the concepts of parallel transport and a connection are introduced and the relation between the Levi-Civita connection and geodesics is elucidated.
Abstract: The mathematics required to analyse higher dimensional curved spaces and space-times is developed in this chapter. General coordinate transformations, tangent spaces, vectors and tensors are described. Lie derivatives and covariant derivatives are motivated and defined. The concepts of parallel transport and a connection is introduced and the relation between the Levi-Civita connection and geodesics is elucidated. Christoffel symbols the Riemann tensor are defined as well as the Ricci tensor, the Ricci scalar and the Einstein tensor, and their algebraic and differential properties are described (though technical details of the derivationa of the Rimeann tensor are let to an appendix).
30 May 2023
TL;DR: In this article , a non-associative Ricci tensor and curvature scalar defined by symmetric metric structures and generalized (non) linear connections are used for defining non-Associative versions of Grigori Perelman F- and W-functionals for Ricci flows and computing associated thermodynamic variables.
Abstract: We extend to a theory of nonassociative geometric flows a string-inspired model of nonassociative gravity determined by star product and R-flux deformations. The nonassociative Ricci tensor and curvature scalar defined by (non) symmetric metric structures and generalized (non) linear connections are used for defining nonassociative versions of Grigori Perelman F- and W-functionals for Ricci flows and computing associated thermodynamic variables. We develop and apply the anholonomic frame and connection deformation method, AFCDM, which allows us to construct exact and parametric solutions describing nonassociative geometric flow evolution scenarios and modified Ricci soliton configurations with quasi-stationary generic off-diagonal metrics. There are provided explicit examples of solutions modelling geometric and statistical thermodynamic evolution on a temperature-like parameter of modified black hole configurations encoding nonassociative star-product and R-flux deformation data. Further perspectives of the paper are motivated by nonassociative off-diagonal geometric flow extensions of the swampland program, related conjectures and claims on geometric and physical properties of new classes of quasi-stationary Ricci flow and black hole solutions.
TL;DR: In this article , local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian are presented.
Abstract: : This article presents new local and global gradient estimates of Li - Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry - Émery Ricci curvature tensor and fi nd utility in proving fairly general Harnack inequalities and Liouville - type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinea - rities already of huge interest and applications. Some consequences are presented and discussed.
29 Apr 2023
TL;DR: In this paper , the Ricci flow out of spaces with edge type conical singularities along a closed, embedded curve is studied and the existence of a Ricci soliton solution is proved.
Abstract: We study the Ricci flow out of spaces with edge type conical singularities along a closed, embedded curve. Under the additional assumption that for each point of the curve, our space is locally modelled on the product of a fixed positively curved cone and a line, we show existence of a solution to Ricci flow $(M,g(t))$ for $t\in (0,T],$ which converges back to the singular space as $t\searrow 0$ in the pointed Gromov-Hausdorff topology. We also prove curvature estimates for the solution and, for edge points, we show that the tangent flow at these points is a positively curved expanding Ricci soliton solution crossed with a line.
TL;DR: In this paper , it was shown that the Ricci flow exists for a short time and that the scalar curvature lower bound is preserved along with the Yamabe invariant.
Abstract: In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon (2002) that the Ricci flow exists for a short time. We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in the distributional sense provided that the initial metric is W1,p for some n < p ⩽ ∞. As an application, we use this result to study the relation between the Yamabe invariant and Ricci flat metrics. We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in the distributional sense, then the manifold is isometric to a Ricci flat manifold.
15 Mar 2023
TL;DR: In this article , it was shown that Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvatures, and that there exists a homogeneous space of dimension $d=8n-4$ with metrics whose Ricci tensor is not $(d-4)-positive.
Abstract: We show that, for any $n\geq 2$, there exists a homogeneous space of dimension $d=8n-4$ with metrics of $\mathrm{Ric}_{\frac{d}{2}-5}>0$ if $n
eq 3$ and $\mathrm{Ric}_6>0$ if $n=3$ which evolve under the Ricci flow to metrics whose Ricci tensor is not $(d-4)$-positive. Consequently, Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature. This extends a theorem of B\"ohm and Wilking in the case of $n=2$.
25 May 2023
TL;DR: In this article , it was shown that a solution of degenerate parabolic complex Monge-Amp\`ere equations starting from arbitrarily positive (1, 1)-currents is smooth outside some analytic subset, generalizing works by Di Nezza-Lu.
Abstract: We study the nature of finite-time singularities for the Chern-Ricci flow, partially answering a question of Tosatti-Weinkove. We show that a solution of degenerate parabolic complex Monge-Amp\`ere equations starting from arbitrarily positive (1,1)-currents are smooth outside some analytic subset, generalizing works by Di Nezza-Lu. We extend Guedj-Lu's recent approach to establish uniform a priori estimates for degenerate complex Monge-Amp\`ere equations on compact Hermitian manifolds. We apply it to studying the Chern-Ricci flows on complex log terminal varieties starting from an arbitrary current.
TL;DR: In this paper , a non-associative Ricci tensor and curvature scalar defined by symmetric metric structures and generalized (non) linear connections are used for defining non-Associative versions of Grigori Perelman F- and W-functionals for Ricci flows and computing associated thermodynamic variables.
Abstract: We extend to a theory of nonassociative geometric flows a string-inspired model of nonassociative gravity determined by star product and R-flux deformations. The nonassociative Ricci tensor and curvature scalar defined by (non) symmetric metric structures and generalized (non) linear connections are used for defining nonassociative versions of Grigori Perelman F- and W-functionals for Ricci flows and computing associated thermodynamic variables. We develop and apply the anholonomic frame and connection deformation method, AFCDM, which allows us to construct exact and parametric solutions describing nonassociative geometric flow evolution scenarios and modified Ricci soliton configurations with quasi-stationary generic off-diagonal metrics. There are provided explicit examples of solutions modelling geometric and statistical thermodynamic evolution on a temperature-like parameter of modified black hole configurations encoding nonassociative star-product and R-flux deformation data. Further perspectives of the paper are motivated by nonassociative off-diagonal geometric flow extensions of the swampland program, related conjectures and claims on geometric and physical properties of new classes of quasi-stationary Ricci flow and black hole solutions.
TL;DR: In this paper , the authors numerically evolve the time-symmetric foliations of a family of spherically symmetric asymptotically flat wormholes under the $1$-loop renormalization group flow of the non-linear $\sigma$-model, the Ricci flow, and under the RG-2 flow.
07 Feb 2023
TL;DR: In this article , Li et al. constructed the Linearly nearly Euclidean (LNE) manifold for DNNs as a background to deal with perturbations by introducing a partial differential equation on metrics, i.e., the Ricci flow, and proved the dynamical stability and convergence of the LNE metric with the $L 2$-norm perturbation.
Abstract: In this paper, we consider Discretized Neural Networks (DNNs) consisting of low-precision weights and activations, which suffer from either infinite or zero gradients due to the non-differentiable discrete function in the training process. In this case, most training-based DNNs employ the standard Straight-Through Estimator (STE) to approximate the gradient w.r.t. discrete values. However, the STE gives rise to the problem of gradient mismatch, due to the perturbations of the approximated gradient. To address this problem, this paper reveals that this mismatch can be viewed as a metric perturbation in a Riemannian manifold through the lens of duality theory. Further, on the basis of the information geometry, we construct the Linearly Nearly Euclidean (LNE) manifold for DNNs as a background to deal with perturbations. By introducing a partial differential equation on metrics, i.e., the Ricci flow, we prove the dynamical stability and convergence of the LNE metric with the $L^2$-norm perturbation. Unlike the previous perturbation theory whose convergence rate is the fractional powers, the metric perturbation under the Ricci flow can be exponentially decayed in the LNE manifold. The experimental results on various datasets demonstrate that our method achieves better and more stable performance for DNNs than other representative training-based methods.
TL;DR: In this paper , the Ricci tensor of an almost pseudo-Ricci symmetric spacetime with Gray's decomposition of the gradient of Ricci's tensor is analyzed.
TL;DR: In this article , the stability and instability of ALE Ricci-flat metrics around which a Łojasiewicz inequality is satisfied for a variation of Perelman's $$\lambda $$ functional adapted to the ALE situation was studied.
Abstract: We study the stability and instability of ALE Ricci-flat metrics around which a Łojasiewicz inequality is satisfied for a variation of Perelman’s $$\lambda $$ functional adapted to the ALE situation and denoted $$\lambda _{{\text {ALE}}}$$ . This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Łojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5.
TL;DR: In this article , a local Sobolev inequality for complete Ricci flows has been shown to depend only on the Nash entropy based on the center of the disk, and consequently depends only on volume of a disk.
08 May 2023
TL;DR: In this article , the authors give a new proof based on inverse mean curvature flow that $(M,g) is either flat or has non-Euclidean volume growth.
Abstract: Let $(M,g)$ be a complete, connected, non-compact Riemannian three-manifold with non-negative Ricci curvature satisfying $Ric\geq\varepsilon\,\operatorname{tr}(Ric)\,g$ for some $\varepsilon>0$. In this note, we give a new proof based on inverse mean curvature flow that $(M,g)$ is either flat or has non-Euclidean volume growth. In conjunction with results of J. Lott and of M.-C. Lee and P. Topping, this gives an alternative proof of a conjecture of R. Hamilton recently proven by A. Deruelle, F. Schulze, and M. Simon using Ricci flow.
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01 Jan 2023
TL;DR: In this article , a geometric averaging technique induced by the Ricci flow is proposed to compare a given (generalized) Einstein initial data set with another distinct Einstein initial set, both supported on a given closed n-dimensional manifold.
Abstract: The goal of these notes is to make accessible to interested readers a case study where two prominent avenues of research of geometric analysis, Ricci flow and Einstein constraints theory, interact in a quite remarkable way. In particular, the theme we discuss and illustrate here is the characterization of a geometric averaging technique, induced by the Ricci flow, that allows us to compare a given (generalized) Einstein initial data set with another distinct Einstein initial data set, both supported on a given closed n-dimensional manifold, i.e., on a compact n-dimensional manifold $$\varSigma $$ without boundary.
16 Jun 2023
TL;DR: In this paper , the authors prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow.
Abstract: In this paper we prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply such estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature.
28 Feb 2023
TL;DR: The generalized Einstein Hilbert action is an extension of the classic scalar curvature energy and Perelman F functional which incorporates a closed three-form as mentioned in this paper , which is known as generalized Ricci solitons, which arise naturally in mathematical physics, complex geometry, and generalized geometry.
Abstract: The generalized Einstein Hilbert action is an extension of the classic scalar curvature energy and Perelman F functional which incorporates a closed three-form. The critical points are known as generalized Ricci solitons, which arise naturally in mathematical physics, complex geometry, and generalized geometry. Through a delicate analysis of the group of generalized gauge transformations, and implementing a novel connection, we give a simple formula for the second variation of this energy which generalizes the Lichnerowicz operator in the Einstein case. As an application, we show that all Bismut flat manifolds are linearly stable critical points, and admit nontrivial deformations arising from Lie theory. Furthermore, this leads to extensions of classic results of Koiso and Podesta, Spiro, Kr\"oncke to the moduli space of generalized Ricci solitons. To finish we classify deformations of the Bismut-flat structure on S3 and show that some are integrable while others are not.
TL;DR: In this paper , complete Type I ancient Ricci flows with positive sectional curvature were studied and the main results were as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the closed case, if the dimension is even, then all such modern Ricci flow must be notcollapsed at all scales.
Abstract: In this paper, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the closed case, if the dimension is even, then all such ancient solutions must be noncollapsed on all scales. This furthermore gives a complete classification for three-dimensional noncompact Type I ancient solutions without assuming the noncollapsing condition.
22 Jun 2023
TL;DR: The stability of a recently developed piecewise flat Ricci flow is investigated, using a linear stability analysis and numerical simulations, and a class of piecewise-flat approximations of smooth manifolds is adapted to avoid an inherent numerical instability as discussed by the authors .
Abstract: The stability of a recently developed piecewise flat Ricci flow is investigated, using a linear stability analysis and numerical simulations, and a class of piecewise flat approximations of smooth manifolds is adapted to avoid an inherent numerical instability. These adaptations have also been used in a related paper to show the convergence of the piecewise flat Ricci flow to known smooth Ricci flow solutions for a variety of manifolds.
TL;DR: In this article , the collapsing of Calabi-Yau metrics and Kähler-Ricci flows on fiber spaces where the base is smooth were studied and an explicit bound for the real codimension-2 Hausdorff measure of the Cheeger-Colding singular set was obtained.
Abstract: Abstract We study the collapsing of Calabi–Yau metrics and of Kähler–Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov–Hausdorff limit of the Kähler–Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension-2 Hausdorff measure of the Cheeger–Colding singular set and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.
20 Jan 2023
TL;DR: In this article , it was shown that the theory of convergence holds naturally on any Ricci flows induced by Ricci shrinkers and that the compactness and curvature boundedness assumptions of the Ricci flow theory hold naturally.
Abstract: This paper is the sequel to our study of heat kernels on Ricci shrinkers in \cite{LW20}. In this paper, we improve many estimates in \cite{LW20} and extend the recent progress of Bamler \cite{Bam20a}. In particular, we drop the compactness and curvature boundedness assumptions and show that the theory of $\IF$-convergence holds naturally on any Ricci flows induced by Ricci shrinkers.