Analysis on free Riemannian path spaces

The gradient operator is defined on the free path space with reference measure Pμ, the law of the Brownian motion on the base manifold with initial distribution μ, where μ has strictly positive density w.r.t. the volume measure. The formula of integration by parts is established for the underlying directional derivatives, which implies the closability of the gradient operator so that it induces a conservative Dirichlet form on the free path space. The log-Sobolev inequality for this Dirichlet form is established and, consequently, the transportation cost inequality is obtained for the associated intrinsic distance.     

A concrete estimate for the weak Poincar�� inequality on loop space

The aim of the paper is to study the pinned Wiener measure on the loop space over a simply connected compact Riemannian manifold together with a Hilbert space structure and the Ornstein?CUhlenbeck operator d*d. We give a concrete estimate for the weak Poincaré inequality, assuming positivity of the Ricci curvature of the underlying manifold. The order of the rate function is s ^??? for any ?? >?0.     

Functional Inequalities and Spectrum Estimates: The Infinite Measure Case

As a continuation of [13] where a Poincaré-type inequality was introduced to study the essential spectrum on the L²-space of a probability measure, this paper provides a modification of this inequality so that the infimum of the essential spectrum is well described even if the reference measure is infinite. High-order eigenvalues as well as the corresponding semigroup are estimated by using this new inequality. Criteria of the inequality and estimates of the inequality constants are presented. Finally, some concrete examples are considered to illustrate the main results. In particular, estimates of high-order eigenvalues obtained in this paper are sharp as checked by two examples on the Euclidean space.     

Pointwise inequalities of logarithmic type in Hardy-Hölder spaces

We prove some optimal logarithmic estimates in the Hardy space H ^∞(G) with Hölder regularity, where G is the open unit disk or an annular domain of ?. These estimates extend the results established by S.Chaabane and I.Feki in the Hardy-Sobolev space H ^k,∞ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem for the Laplace operator and the Robin inverse problem.     

Functional inequality on path space over a non-compact Riemannian manifold

We prove the existence of the O-U Dirichlet form and the damped O-U Dirichlet form on path space over a general non-compact Riemannian manifold which is complete and stochastically complete. We show a weighted log-Sobolev inequality for the O-U Dirichlet form and the (standard) log-Sobolev inequality for the damped O-U Dirichlet form. In particular, the Poincaré inequality (and the super Poincaré inequality) can be established for the O-U Dirichlet form on path space over a class of Riemannian manifolds with unbounded Ricci curvatures. Moreover, we construct a large class of quasi-regular local Dirichlet forms with unbounded random diffusion coefficients on path space over a general non-compact manifold.     

Dimension-Independent Estimates for Heat Operators and Harmonic Functions

We establish dimension-independent estimates related to heat operators e ^tL on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace–Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise bounds on harmonic and subharmonic functions, while in the latter example, we obtain pointwise bounds on harmonic functions when a generalized curvature-dimension inequality is satisfied.     

Evolution of harmonic maps on manifolds flat at infinity

The aim of this article is to prove a global existence result with small data for the heat flow for harmonic maps from a manifold flat at infinity into a compact manifold. By flat at infinity we mean that the growth rate of the volumes of the balls on the manifold is the same as in the flat space. This is true for any manifold for small enough radius, but is in general not true when the radius of the ball grows. So prescribing such a growth rate also at infinity selects a class of manifolds on which our result holds. In this setting estimates are available for the heat kernel and its gradient on the base manifold. From such estimates it is easy to get L ^p −L ^q bounds for the heat kernel. A contraction principle argument then yields a local existence result in a suitable Sobolev space and a global existence result for small data.     

Spin c geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds

From the existence of parallel spinor fields on Calabi-Yau, hyper-Kähler or complex flat manifolds, we deduce the existence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the submanifolds are compact, we obtain sharp estimates on their Betti numbers which generalize those obtained by Smoczyk in [49]. When the ambient manifold is Kähler-Einstein with positive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of Kählerian Killing spinor fields for some particular spin ^c structures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. These results also generalize some theorems by Smoczyk in [50]. Finally, applications on the Morse index of minimal Lagrangian submanifolds are obtained.     

Special Slant Surfaces and a Basic Inequality

A slant immersion is an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. A slant submanifold is called proper if it is neither holomorphic nor totally real. In [2], the author proved that, for any proper slant surface M with slant angle θ in a complex-space-form $?detilde M^2(4?silon)$ with constant holomorphic sectional curvature 4?, the squared mean curvature and the Gauss curvature of M satisfy the following basic inequality: H²(p) ≥ 2K(p) ? 2(1 + 3 cos²θ)?. Every proper slant surface satisfying the equality case of this inequality is special slant. One purpose of this article is to completely classify proper slant surfaces which satisfy the equality case of this inequality. Another purpose of this article is to completely classify special slant surfaces with constant mean curvature. Further results on special slant surfaces are also presented.     

Positively oriented ideal triangulations on hyperbolic three-manifolds

Let M³ be a non-compact hyperbolic 3-manifold that has a triangulation by positively oriented ideal tetrahedra. We show that the gluing variety defined by the gluing consistency equations is a smooth complex manifold with dimension equal to the number of boundary components of M³. Moreover, we show that the complex lengths of any collection of non-trivial boundary curves, one from each boundary component, give a local holomorphic parameterization of the gluing variety. As an application, some estimates for the size of hyperbolic Dehn surgery space of once-punctured torus bundles are given.     

Volume Growth,Number of Ends,and the Topology of a Complete Submanifold

Given a complete isometric immersion φ:P ^m ?N ⁿ in an ambient Riemannian manifold N ⁿ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space  \(M^{n}_{w}\) , we determine a set of conditions on the extrinsic curvatures of P that guarantee that the immersion is proper and that P has finite topology in line with the results reported in Bessa et al. (Commun. Anal. Geom. 15(4):725–732, 2007) and Bessa and Costa (Glasg. Math. J. 51:669–680, 2009). When the ambient manifold is a radially symmetric space, an inequality is shown between the (extrinsic) volume growth of a complete and minimal submanifold and its number of ends, which generalizes the classical inequality stated in Anderson (Preprint IHES, 1984) for complete and minimal submanifolds in ? ⁿ . As a corollary we obtain the corresponding inequality between the (extrinsic) volume growth and the number of ends of a complete and minimal submanifold in hyperbolic space, together with Bernstein-type results for such submanifolds in Euclidean and hyperbolic spaces, in the manner of the work Kasue and Sugahara (Osaka J. Math. 24:679–704, 1987).     

The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures

We state and prove a Chern–Osserman-type inequality in terms of the volume growth for minimal surfaces S which have finite total extrinsic curvature and are properly immersed in a Cartan–Hadamard manifold N with sectional curvatures bounded from above by a negative quantity K _N ≤b<0 and such that they are not too curved (on average) with respect to the hyperbolic space with constant sectional curvature given by the upper bound b. We also prove the same Chern–Osserman-type inequality for minimal surfaces with finite total extrinsic curvature and properly immersed in an asymptotically hyperbolic Cartan–Hadamard manifold N with sectional curvatures bounded from above by a negative quantity K _N ≤b<0.     

The Sharp Lower Bound of the First Eigenvalue of the Sub-Laplacian on a Quaternionic Contact Manifold

The main technical result of the paper is a Bochner type formula for the sub-Laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz’s theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(n)Sp(1) components of the qc-Ricci curvature. It is shown that in the case of a 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is a round 3-Sasakian sphere. Another goal of the paper is to establish a priori estimates for square integrals of horizontal derivatives of smooth compactly supported functions. As an application, we prove a sharp inequality bounding the horizontal Hessian of a function by its sub-Laplacian on the quaternionic Heisenberg group.     

On the mean curvature of semi-Riemannian graphs in semi-Riemannian warped products

We investigate the mean curvature of semi-Riemannian graphs in the semi-Riemannian warped product M× _f ? _?? , where M is a semi-Riemannian manifold, ? _?? is the real line ? with metric ??dt ² (???=?±1), and f: M???^?+? is the warping function. We obtain an integral formula for mean curvature and some results dealing with estimates of mean curvature, among these results is a Heinz?CChern type inequality.     

Heat flow and Hardy inequality in complete Riemannian manifolds with singular initial conditions

Upper bounds are obtained for the heat content of an open set D with singular initial condition f on a complete Riemannian manifold, provided (i) the Dirichlet-Laplace-Beltrami operator satisfies a strong Hardy inequality, and (ii) f satisfies an integrability condition. Precise asymptotic results for the heat content are obtained for an open bounded and connected set D in Euclidean space with C² boundary, and with initial condition f(x)=δ(x)^−α,0<α<2, where δ(x) is the distance from x to the boundary of D.     

Brézis-Gallouët-Wainger type inequality with a double logarithmic term in the Hölder space: Its sharp constants and extremal functions

We investigate the sharp constants in a Brézis-Gallouët-Wainger type inequality with a double logarithmic term in the Hölder space in a bounded domain in Rⁿ. Ibrahim, Majdoub and Masmoudi gave the sharp constant in the two-dimensional case. We make precise estimates to give the sharp constants, and pass to the case of higher dimensions n≥2. We can also show that the inequality with fixed constants including the sharp ones admits an extremal function under a suitable condition when the domain is a ball.     

The metric geometry of the manifold of Riemannian metrics over a closed manifold

We prove that the L ² Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L ² metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.     

A Riemannian interpolation inequality à la Borell, Brascamp and Lieb

A concavity estimate is derived for interpolations between L ¹(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian versions of these theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature. The method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal mass transport and interpolating maps on a Riemannian manifold. Oblatum 1-XII-2000 & 3-V-2001?Published online: 13 August 2001     

The symplectic normal space of a cotangent-lifted action

For the cotangent bundle T^∗Q of a smooth Riemannian manifold acted upon by the lift of a smooth and proper action by isometries of a Lie group, we characterize the symplectic normal space at any point. We show that this space splits as the direct sum of the cotangent bundle of a linear space and a symplectic linear space coming from reduction of a coadjoint orbit. This characterization of the symplectic normal space can be expressed solely in terms of the group action on the base manifold and the coadjoint representation. Some relevant particular cases are explored.     

Semi-Parallel, Semi-Symmetric Immersions and Chen’s Equality

Recently, B. Y. Chen introduced a new invariant δ(n₁,n₂,…,n_k) of a Riemannian manifold and proved a basic inequality between the invariant and the extrinsic invariant if, where H is the mean curvature of an immersion Mⁿ in a real space form R^m(ε) of constant curvature ε. He pointed out that such inequality also holds for a totally real immersion in a complex space form. The immersion is called ideal (by B. Y. Chen) if it satisfies the equality case of such inequality identically. In this paper we classify ideal semi-parallel immersions in an Euclidean space if their normal bundle is flat, and prove that every ideal semi-parallel Lagrangian immersion in a complex space form is totally geodesic, moreover this result also holds for ideal semi-symmetric Lagrangian immersions in complex projective space and hyperbolic space.