Gradient estimates for a weighted heat equation under weighted Yamabe flow and applications

ABSTRACT In this paper, we introduce a new class of doubly nonlinear parabolic equations, motivated by the ‐Sobolev flow problem and the ‐Hilfer operators in fractional Sobolev space. We discuss some special cases; in particular, when in a bounded domain in Euclidean space, we obtain the fractional Yamabe flow, and when and , we obtain the Yamabe flow. We investigate the existence of a weak solution to these new problems using the Galerkin method and discuss other results throughout the work.

Abstract This paper mainly uses the Harnack expression (HE) of the Yamabe flow to study the eternal solutions to the locally conformally flat Yamabe flow. First, we give the HE of the Yamabe flow, and we calculate the evolution of this Harnack expression (EHE). Then, we use this EHE to get the main theorem.

Abstract In this work we introduce a family of conformal flows generalizing the classical Yamabe flow. We prove that for a large class of such flows long-time existence holds, and the arguments are in fact simpler than in the classical case. Moreover, we establish convergence for the case of negative scalar curvature and expect a similar statement for the positive and the flat cases as well.

Extending CR Yamabe flow and Yamabe flow with boundary

Abstract We define the hyperbolic Yamabe flow and obtain some properties of its stationary solutions, namely, of hyperbolic Yamabe solitons. We consider immersed submanifolds as hyperbolic Yamabe solitons and prove that, under certain assumptions, a hyperbolic Yamabe soliton hypersurface is a pseudosymmetric or a metallic shaped hypersurface. We characterize the hyperbolic Yamabe soliton factor manifolds of a multiply twisted, multiply warped, doubly warped, and warped product manifold and provide a classification for a complete gradient hyperbolic Yamabe soliton factor manifold. We also determine the conditions for the factor manifolds to be hyperbolic Yamabe solitons if the manifold is a hyperbolic Yamabe soliton and illustrate this result for a physical model of the universe, namely, for the Robertson–Walker spacetime.

Yamabe flow and locally conformally flat manifolds with positive pinched Ricci curvature

Abstract Let ( M , ∂ M , g ) \left(M,\partial M,g) be a compact Riemannian manifold with boundary. As a generalization of the Yamabe invariant with boundary Y ( M , ∂ M , g ) Y\left(M,\partial M,g) , we define the kth Yamabe invariant with boundary Y k ( M , ∂ M , g ) {Y}_{k}\left(M,\partial M,g) . We prove some of its properties and study when it can be attained by the generalized metric. We also prove a version of conformal Schwarz lemma on ( M , ∂ M , g ) \left(M,\partial M,g) by using the Yamabe flow with boundary.

In the year 2019, Guler and Crasmareanu [6] conducted an investigation into another geometric flow known as the Ricci-Yamabe map. This map is nothing but a scalar combination of the Ricci and the Yamabe flow [7]. The primary objective of the current paper is to provide a characterization of a Ricci Yamabe soliton on a para-Sasakian manifold [17]. To commence, we prove that a para-Sasakian manifold admits a nearly quasi-Einstein manifold. Moreover, we discuss whether such a manifold is shrinking, expanding, or steady. Subsequently, we generalize these findings to Ricci-Yamabe solitons on para-Sasakian manifolds equipped with a quarter symmetric metric connection. Lastly, we furnish an illustration of a three-dimensional para-Sasakian manifold admitting a Ricci-Yamabe soliton which satisfies our results.

A Dual Yamabe Flow and Related Integral Flows

Abstract In this paper, we investigate the structure of certain solutions of the fully nonlinear Yamabe flow, which we call almost quotient Yamabe solitons as they extend quite naturally those already called quotient Yamabe solitons. We present sufficient conditions for a compact almost quotient Yamabe soliton to be either trivial or isometric with an Euclidean sphere. We also characterize noncompact almost gradient quotient Yamabe solitons satisfying certain conditions on both its Ricci tensor and potential function.

First eigenvalues evolution for some geometric operators along the Yamabe flow

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

In this work, we study the convergence of the normalized Yamabe flow with positive Yamabe constant on a class of pseudo-manifolds that includes stratified spaces with iterated cone-edge metrics. We establish convergence under a low-energy condition. We also prove a concentration-compactness dichotomy, and investigate what the alternatives to convergence are. We end by investigating a non-convergent example due to Viaclovsky in more detail.

Abstract We study the Yamabe flow starting from an asymptotically flat manifold ( M n , g 0 ) (M^{n},g_{0}) . We show that the flow converges to an asymptotically flat, scalar flat metric in a weighted global sense if Y ⁢ ( M , [ g 0 ] ) > 0 Y(M,[g_{0}])>0 , and show that the flow does not converge otherwise. If the scalar curvature is nonnegative and integrable, then the ADM mass at time infinity drops by the limit of the total scalar curvature along the flow.

In this work we establish long-time existence of the normalized Yamabe flow with positive Yamabe constant on a class of manifolds that includes spaces with incomplete cone-edge singularities. We formulate our results axiomatically, so that our results extend to general stratified spaces as well, provided certain parabolic Schauder estimates hold. The central analytic tool is a parabolic Moser iteration, which yields uniform upper and lower bounds on both the solution and the scalar curvature.

Abstract The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) , the weighted Yamabe problem consists on finding another smooth metric measure space conformal to ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ + μ e − ϕ ∕ m \lambda +\mu {e}^{-\phi /m} for some constants μ \mu and λ \lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . On the other hand, we prove a version of the conformal Schwarz lemma on ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.

Normalized Yamabe flow on manifolds with bounded geometry

Chow and Luo [1] in 2003 had shown that the combinatorial analogue of the Hamilton Ricci flow on surfaces under certain conditions converges to Thruston?s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. Crucial assumption in the paper [1] was that the weights are nonnegative. Recently we have shown that same statement on convergence can be proved under weaker condition: some weights can be negative and should satisfy certain inequalities, [3]. Moreover, in [6] notions of degenerate circle packing and corresponding metric were introduced. In [6] theory of combinatorial Ricci flow for such metrics was developed, which includes Chow-Luo theory as a partial case for nondegenerate circle packing and nonnegative weights on edges. On the other hand, in [2] the combinatorial Yamabe flow was introduced and investigated. In [7, 8] we developed weighted modification of Yamabe flow. In this paper we merge ideas from these two theories and introduce weighted combinatorial Ricci flow on metrics defined by degenerate circle packings. We prove that under certain conditions for any initial metric the flow converges to a unique metric of constant curvature.

The weighted Yamabe flow with boundary