Blaschke's rolling theorem for manifolds¶with boundary

For a complete Riemannian manifold M with compact boundary ∂M denote by $\Cut$ the cut locus of $\f M$ in M. The rolling radius of M is roll(M)≔ dist(∂M, ?_{∂ M} ). Let Focal(∂M) be the focal distance of ∂M in M. Then conditions are given that imply the equality roll(M)= Focal(∂M). This generalizes Blaschke's rolling theorem from bounded convex domains in Euclidean space to more general Euclidean domains and to Riemannian manifolds with boundary. Received: 28 August 1998 / Revised version: 8 February 1999     

A global rigidity theorem for weakly Landsberg manifolds

In this paper we study a global rigidity property for weakly Landsberg manifolds and prove that a closed weakly Landsberg manifold with the negative flag curvature must be Riemannian.     

Some comparison theorems for K?hler manifolds

In this work, we will verify some comparison results on K?hler manifolds. They are: complex Hessian comparison for the distance function from a closed complex submanifold of a K?hler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scalar curvature. This work is motivated by comparison results of Li and Wang (J Differ Geom 69(1):43–47, 2005).     

Comparison and vanishing theorems for Kähler manifolds

In this paper, we consider orthogonal Ricci curvature \(Ric^{\perp }\) for Kähler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate subtle relationship among them. As a consequence of the vanishing theorem, we show that any compact Kähler manifold with positive orthogonal Ricci curvature must be projective. This result complements a recent result of Yang (RC-positivity, rational connectedness, and Yau’s conjecture. arXiv:1708.06713) on the projectivity under the positivity of holomorphic sectional curvature. The simply-connectedness is shown when the complex dimension is smaller than five. Further study of compact Kähler manifolds with \(Ric^{\perp }>0\) is carried in Ni et al. (Manifolds with positive orthogonal Ricci curvature. arXiv:1806.10233).     

A Frölicher-type inequality for generalized complex manifolds

We prove a Frölicher-type inequality for a compact generalized complex manifold \(M\), and show that the equality holds if and only if \(M\) satisfies the generalized \(\partial \bar{\partial }\)-Lemma. In particular, this gives a unified proof of analogous results in the complex and symplectic cases.     

The Gauss–Bonnet–Chern mass for graphic manifolds

In this paper, we prove a positive mass theorem and Penrose-type inequality of the Gauss–Bonnet–Chern mass $m_2$ for the graphic manifold with flat normal bundle.     

Vanishing theorems for some classes of Riemann–Cartan manifolds

In this paper, a classification of Riemann–Cartan manifolds based on the orthogonal decomposition of the torsion tensor is given. Problems on the existence of two classes ℘₁ ⨁ ℘₂ and ℘₃ of Riemann–Cartan spaces are discussed.     

Gauss–Newton method for convex composite optimizations on Riemannian manifolds

A notion of quasi-regularity is extended for the inclusion problem ${F(p)\in C}$ , where F is a differentiable mapping from a Riemannian manifold M to ${\mathbb R^n}$ . When C is the set of minimum points of a convex real-valued function h on ${\mathbb R^n}$ and DF satisfies the L-average Lipschitz condition, we use the majorizing function technique to establish the semi-local convergence of sequences generated by the Gauss-Newton method (with quasi-regular initial points) for the convex composite function h ? F on Riemannian manifold. Two applications are provided: one is for the case of regularities on Riemannian manifolds and the other is for the case when C is a cone and DF(p ₀)(·) ? C is surjective. In particular, the results obtained in this paper extend the corresponding one in Wang et?al. (Taiwanese J Math 13:633?C656, 2009).     

Blow-up solutions for Hardy–Sobolev equations on compact Riemannian manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension \(n\ge 6\), \(\xi _0\in M\), and we are concerned with the following Hardy–Sobolev elliptic equations:

$$\begin{aligned} -\Delta _gu+h(x)u=\frac{u^{2^{*}(s)-1-\epsilon }}{d_{g}(x,\xi _0)^s},\ \ \ \ u>0\ \ \mathrm{in} \ \ M, \end{aligned}$$

(0.1)

where \(\Delta _g\,=\,\mathrm{div}_g(\nabla )\) is the Laplace–Beltrami operator on M, h(x) is a \(C^1\) function on M, \(\epsilon \) is a sufficiently small real parameter, \(2^{*}(s):=\frac{2(n-s)}{n-2}\) is the critical Hardy–Sobolev exponent with \(s\in (0,2)\), and \(d_{g}\) is the Riemannian distance on M. Performing the Lyapunov–Schmidt reduction procedure, we obtain the existence of blow-up families of positive solutions of problem (0.1).

     

Enumerative meaning of mirror maps for toric Calabi–Yau manifolds

We prove that the inverse of a mirror map for a toric Calabi–Yau manifold of the form _KY$K_Y$, where Y$Y$ is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov–Witten invariants defined by Fukaya–Oh–Ohta–Ono (2010)  [15]. Such a relation between mirror maps and disk counting invariants was first conjectured by Gross and Siebert (2011)  [24, Conjecture 0.2 and Remark 5.1] as part of their program, and was later formulated in terms of Fukaya–Oh–Ohta–Ono’s invariants in the toric Calabi–Yau case in Chan et al. (2012)  [8, Conjecture 1.1].     

Generalized Liouville property for Schrödinger operator on Riemannian manifolds

In this paper, we prove that the dimension of the space of positive (bounded, respectively) -harmonic functions on a complete Riemannian manifold with -regular ends is equal to the number of ends (-nonparabolic ends, respectively). This result is a solution of an open problem of Grigor'yan related to the Liouville property for the Schr?dinger operator . We also prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincaré inequality and the finite covering condition on each end, then the dimension of the space of positive (bounded, respectively,) solutions for the Schr?dinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends (-nonparabolic ends, respectively). This is a partial answer of the question, suggested by Li, related to the regularity of ends of a complete Riemannian manifold. Especially, our results directly generalize various earlier results of Yau, of Li and Tam, of Grigor'yan, and of present authors, but with different techniques that the peculiarity of the Schr?dinger operator demands. Received: 4 April 2000; in final form: 19 September 2000 / Published online: 25 June 2001     

The Bridgeman–Kahn identity for hyperbolic manifolds with cusped boundary

In this note, we extend the Bridgeman–Kahn identity to all finite-volume orientable hyperbolic n-manifolds with totally geodesic boundary. In the compact case, Bridgeman and Kahn are able to express the manifold’s volume as the sum of a function over only the orthospectrum. For manifolds with non-compact boundary, our extension adds terms corresponding to intrinsic invariants of boundary cusps.     

Gallot–Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications

We extend the Gallot–Tanno theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over a manifold admits a parallel symmetric (0, 2)-tensor then it is Riemannian. Applications of this result to the existence of metrics with distinct Levi-Civita connections but having the same unparametrized geodesics and to the projective Obata conjecture are given. We also apply our result to show that the holonomy group of a closed (O(p + 1, q), S ^p,q )-manifold does not preserve any nondegenerate splitting of \mathbb R^p+1,q{\mathbb {R}^{p+1,q}}.     

Korpelevich’s method for variational inequality problems on Hadamard manifolds

The concept of pseudomonotone vector field on Hadamard manifold is introduced. A variant of Korpelevich??s method for solving the variational inequality problem is extended from Euclidean spaces to constant curvature Hadamard manifolds. Under a pseudomonotone assumption on the underlying vector field, we prove that the sequence generated by the method converges to a solution of variational inequality, whenever it exists. Moreover, we give an example to show the effectiveness of our method.     

Malgrange’s vanishing theorem for weakly pseudoconcave C R manifolds

We prove the Malgrange vanishing and approximation theorems for smooth, abstract, weakly pseudoconcave C R manifolds of finite kind.     

Sard’s theorem for mappings between Fréchet manifolds

We prove an infinite-dimensional version of Sard’s theorem for Fréchet manifolds. Let M (respectively, N) be a bounded Fréchet manifold with compatible metric d _M (respectively, d _N ) modeled on Fréchet spaces E (respectively, F) with standard metrics. Let f : M → N be an MC ^k -Lipschitz–Fredholm map with k > max{Ind f, 0}: Then the set of regular values of f is residual in N.     

Multilinear estimates for the Laplace spectral projectors on compact manifolds

The purpose of this Note is to extend to any space dimension the bilinear estimate for eigenfunctions of the Laplace operator on a compact manifold (without boundary) obtained by the authors (preprint: http://www.arxiv.org/abs/math/0308214) in dimension 2. We also give some related trilinear estimates. To cite this article: N. Burq et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).