Mathematics

Let λ(t) be the first eigenvalue of ?��?aR(a>0) under the backward Ricci flow on locally homogeneous 3-manifolds, where R is the scalar curvature. In the Bianchi case, we get the upper and lower bounds of λ(t) . In particular, we show that when the the backward Ricci flow converges to a sub-Riemannian geometry after a proper re-scaling, λ + (t) approaches zero, where λ + (t)=max{λ(t),0} .

In this paper, we study the doubly warped product manifolds with semisymmetric metric connection. We derive the curvatures formulas for doubly warped product manifold with semi-symmetric metric connection in terms of curvatures of components of doubly warped product manifolds. We also prove the necessary and sufficient condition for a doubly warped product manifold to be a warped product manifold. We obtain some results for Einstein doubly warped product manifold and Einstein-like doubly warped product manifold of class A with respect to a semi-symmetric metric connection.

In this article, we construct non-compact complete Einstein metrics on two infinite series of manifolds. The first series of manifolds are vector bundles with S 4m+3 as principal orbit and HP m as singular orbit. The second series of manifolds are R 4m+4 with the same principal orbit. For each case, a continuous 1-parameter family of complete Ricci-flat metrics and a continuous 2-parameter family of complete negative Einstein metrics are constructed. In particular, Spin(7) metrics A 8 and B 8 discovered by Cvetič et al. in 2004 are recovered in the Ricci-flat family. A Ricci flat metric with conical singularity is also constructed on R 4m+4 . Asymptotic limits of all Einstein metrics constructed are studied. Most of the Ricci-flat metrics are asymptotically locally conical (ALC). Asymptotically conical (AC) metrics are found on the boundary of the Ricci-flat family. All the negative Einstein metrics constructed are asymptotically hyperbolic (AH).

A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below 8? . Combining such a strict inequality with previous works by Keller-Mondino-Rivière and Mondino-Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution of the isoperimetric-constrained Willmore problem for tori. Moreover, we deduce the existence of smoothly embedded tori minimising the Helfrich functional with small spontaneous curvature. Furthermore, because of their symmetry, the Delaunay tori can be used to construct spheres of high isoperimetric ratio, leading to an alternative proof of the known result for the genus zero case.

We prove an ϵ -regularity result for the tracefree curvature of a Willmore surface with bounded second fundamental form. For such a surface, we obtain a pointwise control of the tracefree second fundamental form from a small control of its L 2 -norm.Several applications are investigated. Notably, we derive a gap statement for surfaces of the aforementioned type. We further apply our results to deduce regularity results for conformal minimal spacelike immersions into the de Sitter space S 4,1 .

In this paper we make a detailed analysis of conservation principles in the context of a family of fourth-order gravitational theories generated via a quadratic Lagrangian. In particular, we focus on the associated notion of energy and start a program related to its study. We also exhibit examples of solutions which provide intuitions about this notion of energy which allows us to interpret it, and introduce several study cases where its analysis seems tractable. Finally, positive energy theorems are presented in restricted situations.

This is the first in a series of two papers studying mu-cscK metrics and muK-stability, from a new perspective evoked from observations in arXiv:2004.06393 and in this first article. The first paper is about a characterization of mu-cscK metrics in terms of Perelman's W-entropy W ? λ . We regard Perelman's W-entropy as a functional on the tangent bundle TH(X,L) of the space H(X,L) of K"ahler metrics in a given K"ahler class L . The critical points of W ? λ turn out to be μ λ -cscK metrics. When λ?? , the supremum along the fibres gives a smooth functional on H(X,L) , which we call mu-entropy. Then μ λ -cscK metrics are also characterized as critical points of this functional, similarly as extremal metric is characterized as the critical points of Calabi functional. We also prove the W-entropy is monotonic along geodesics, following Berman--Berndtsson's subharmonicity argument. Studying the limit of the W-entropy, we obtain a lower bound of the mu-entropy. This bound is not just analogous, but indeed related to Donaldson's lower bound on Calabi functional by the extremal limit λ?��???.

In this paper we establish new geometric and analytic bounds for Ricci flows, which will form the basis of a compactness, partial regularity and structure theory for Ricci flows in [Bam20a, Bam20b]. The bounds are optimal up to a constant that only depends on the dimension and possibly a lower scalar curvature bound. In the special case in which the flow consists of Einstein metrics, these bounds agree with the optimal bounds for spaces with Ricci curvature bounded from below. Moreover, our bounds are local in the sense that if a bound depends on the collapsedness of the underlying flow, then we are able to quantify this dependence using the pointed Nash entropy based only at the point in question. Among other things, we will show the following bounds: Upper and lower volume bounds for distance balls, dependence of the pointed Nash entropy on its basepoint in space and time, pointwise upper Gaussian bound on the heat kernel and a bound on its derivative and an L 1 -Poincaré inequality. The proofs of these bounds will, in part, rely on a monotonicity formula for a notion, called variance of conjugate heat kernels. We will also derive estimates concerning the dependence of the pointed Nash entropy on its basepoint, which are asymptotically optimal. These will allow us to show that points in spacetime that are nearby in a certain sense have comparable pointed Nash entropy. Hence the pointed Nash entropy is a good quantity to measure local collapsedness of a Ricci flow Our results imply a local ε -regularity theorem, improving a result of Hein and Naber. Some of our results also hold for super Ricci flows.

Let K=R or C . We study basic invariants of submanifolds of solutions M={y=Q(x,a,b)}={b=P(a,x,y)} in coordinates x??K n�? , y?�K , a??K m�? , b?�K under split-diffeomorphisms (x,y,a,b)??f(x,y),g(x,y),?(a,b),?(a,b)) . Two Levi forms exist, and have the same rank r⩽min(n,m) . If M is k -nondegenerate with respect to parameters and l -nondegenerate with respect to variables, Aut(M) is a local Lie group of dimension: dimAut(M)�? n+1+2k+2l 2k+2l )min{(n+1),(m+1)}. Mainly, our goal is to set up foundational material addressed to CR geometers. We focus on n=m=2 , assuming r=1 . In coordinates (x,y,z,a,b,c) , a local equation is: z=c+xa+βxxb+ β ????yaa+c O x,y,a,b (2)+ O x,y,a,b,c (4), with β and β ????representing the two 2 -nondegeneracy invariants at 0 . The associated para-CR PDE system: z y =F(x,y,z, z x , z xx ) & z xxx =H(x,y,z, z x , z xx ), satisfies F z xx ?? from Levi degeneracy. We show in details that the hypothesis of 2 -nondegeneracy with respect to variables is equivalent to F z x z x ?? . This gives CR-geometric meaning to the first two para-CR relative differential invariants encountered independently in arxiv.org/abs/2003.08166/.

In this article, we study several closely related invariants associated to Dirac operators on odd-dimensional manifolds with boundary with an action of the compact group H of isometries. In particular, the equality between equivariant winding numbers, equivariant spectral flow, and equivariant Maslov indices is established. We also study equivariant η -invariants which play a fundamental role in the equivariant analog of Getzler's spectral flow formula. As a consequence, we establish a relation between equivariant η -invariants and equivariant Maslov triple indices in the splitting of manifolds.