Addenda to “The entropy formula for linear heat equation”

We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ ⁿ .In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s entropy in the case of Riemann surfaces.     

Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

In this paper, we study Perelman’s W{{\mathcal W}} -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula for the W{\mathcal{W}} -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions. In particular, we prove a monotonicity theorem and a rigidity theorem for the W{{\mathcal W}} -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W{\mathcal{W}} -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact Riemannian manifolds.     

Invariant manifolds for differential equations

In this paper we discuss the concept ‘generalized exponential dichotomy’ and give the existence ofC ^k invariant manifolds for abstract nonautonomous differential equations in Banach or Hilbert spaces. Also we give a classification of invariant manifolds and an estimate of the locality radius of invariant manifolds.     

On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.     

Applications of Klee’s Dehn–Sommerville Relations

We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations and strengthen Kalai’s result on the number of their edges. I. Novik research partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748. E. Swartz research partially supported by NSF grant DMS-0600502.     

Non-Kähler manifolds and GIT-quotients

Bosio generalized the construction by López de Medrano-Verjovsky- Meersseman (LVM) of a family of non-algebraic compact complex manifolds of any dimension. We describe how to construct the generalized family from certain Geometric Invariant Theory (GIT) quotients. We show that Bosio’s generalization parallels exactly the extension from Mumford’s GIT to the more general GIT developed by Białynicki-Birula and Świȩcicka. This point of view yields new results on the geometry of LVM and Bosio’s manifolds.     

Finsler interpolation inequalities

We extend Cordero-Erausquin et al.’s Riemannian Borell–Brascamp–Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvature-dimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest.     

Convergence of Newton’s Method for Sections on Riemannian Manifolds

The present paper is concerned with the convergence problems of Newton’s method and the uniqueness problems of singular points for sections on Riemannian manifolds. Suppose that the covariant derivative of the sections satisfies the generalized Lipschitz condition. The convergence balls of Newton’s method and the uniqueness balls of singular points are estimated. Some applications to special cases, which include the Kantorovich’s condition and the γ-condition, as well as the Smale’s γ-theory for sections on Riemannian manifolds, are given. In particular, the estimates here are completely independent of the sectional curvature of the underlying Riemannian manifold and improve significantly the corresponding ones due to Dedieu, Priouret and Malajovich (IMA J. Numer. Anal. 23:395–419, 2003), as well as the ones in Li and Wang (Sci. China Ser. A. 48(11):1465–1478, 2005).     

Time-Extremal Navigation in Arbitrary Winds on Conformally Flat Riemannian Manifolds

Journal of Optimization Theory and Applications - This paper aims at solving Zermelo’s navigation problem on conformally flat Riemannian manifolds admitting a ship’s variable...     

Symplectic method for the construction of ergodic measures on invariant submanifolds of nonautonomous hamiltonian systems: Lagrangian manifolds, their structure, and mather homologies

We develop a new approach to the study of properties of ergodic measures for nonautonomous periodic Hamiltonian flows on symplectic manifolds, which are used in many problems of mechanics and mathematical physics. Using Mather’s results on homologies of invariant probability measures that minimize some Lagrangian functionals and the symplectic theory developed by Floer and others for the investigation of symplectic actions and transversal intersections of Lagrangian manifolds, we propose an analog of a Mather-type β-function for the study of ergodic measures associated with nonautonomous Hamiltonian systems on weakly exact symplectic manifolds. Within the framework of the Gromov-Salamon-Zehnder elliptic methods in symplectic geometry, we establish some results on stable and unstable manifolds for hyperbolic invariant sets, which are used in the theory of adiabatic invariants of slowly perturbed integrable Hamiltonian systems. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 675–691, May, 2006.     

Proper Submanifolds in Product Manifolds

The authors obtain various versions of the Omori-Yau's maximum principle on complete properly immersed submanifolds with controlled mean curvature in certain product manifolds,in complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay,and also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature.Using the generalized maximum principle,an estimate on the mean curvature of properly immersed submanifolds with bounded projection in N1 in the product manifold N1 ×N2 is given.Other applications of the generalized maximum principle are also given.     

A note on Jensen’s inequality for BSDEs

Under the Lipschitz assumption and square integrable assumption on g, Jiang proved that Jensen＇s inequality for BSDEs with generator g holds in general if and only if g is independent of y, g is super homogenous in z and g（t, 0） = 0, a.s., a.e.. In this paper, based on Jiang＇s results, under the same assumptions as Jiang＇s, we investigate the necessary and sufficient condition on g under which Jensen＇s inequality for BSDEs with generator g holds for some specific convex functions, which generalizes some known results on Jensen＇s inequality for BSDEs.     

Hyperbolic conservation laws on manifolds. An error estimate for finite volume schemes

Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm is of order h1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch＇s theory which was originally developed in the Euclidian setting. We extend the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.     

Local embeddability of CR manifolds into spheres

In this paper we solve local CR embeddability problem of smooth CR manifolds into spheres under a certain nondegeneracy condition on the Chern–Moser’s curvature tensor. We state necessary and sufficient conditions for the existence of CR embeddings as finite number of equations and rank conditions on the Chern–Moser’s curvature tensors and their derivatives. We also discuss the rigidity of those embeddings. J.-W. Oh was partially supported by BK21-Yonsei University.     

Bordism classes represented by multiple point manifolds of immersed manifolds

We present a geometrical version of Herbert’s theorem determining the homology classes represented by the multiple point manifolds of a self-transverse immersion. Herbert’s theorem and generalizations can readily be read off from this result. The simple geometrical proof is based on ideas in Herbert’s paper. We also describe the relationship between this theorem and the homotopy theory of Thom spaces.     

Division theorems and twisted complexes

We prove new Skoda-type division, or ideal membership, theorems. We work in a geometric setting of line bundles over Kahler manifolds that are Stein away from an analytic subvariety. (This includes complex projective manifolds.) Our approach is to combine the twisted Bochner-Kodaira Identity, used in the Ohsawa-Takegoshi Theorem, with Skoda’s basic estimate for the division problem. Techniques developed by McNeal and the author are then used to provide many examples of new division theorems. Among other applications, we give a modification of a recent result of Siu regarding effective finite generation of certain section rings. Partially supported by NSF grant DMS-0400909.     

Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

In 1993,Tsal proved that a proper holomorphic mapping f:Ω→Ω' from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ' is necessarily totally geodesic provided that r':=rank(Ω')≤rank(Ω):= r,proving a conjecture of the author's motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.     

Note on a theorem of Bangert

We generalize Bangert’s non-hyperbolicity result for uniformly tamed almost complex structures on standard symplectic ℝ²ⁿ to asymptotically standard symplectic manifolds.     

Minimizing weak solutions for calabi’s extremal metrics on toric manifolds

In this paper, we discuss Donaldson’s version of the modified K-energy associated to the Calabi’s extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified K-energy. The second author was partially supported by NSF10425102 in China and the Huo Y-T Fund.     

Statistical manifolds are statistical models

In this note we prove that any smooth (C¹ resp.) statistical manifold can be embedded into the space of probability measures on a finite set. As a result, we get positive answers to Lauritzen’s question and Amari’s question on a realization of smooth (C¹ resp.) statistical manifolds as finite dimensional statistical models.