Mean distance of Brownian motion on a Riemannian manifold

Consider the mean distance of Brownian motion on Riemannian manifolds. We obtain the first three terms of the asymptotic expansion of the mean distance by means of stochastic differential equation for Brownian motion on Riemannian manifold. This method proves to be much simpler for further expansion than the methods developed by Liao and Zheng (Ann. Probab. 23(1) (1995) 173). Our expansion gives the same characterizations as the mean exit time from a small geodesic ball with regard to Euclidean space and the rank 1 symmetric spaces.     

Some limit theorems of killed Brownian motion

In this paper, we prove some limit theorems for killed Brownian motion during its life time. The emphases are on quasi-stationarity and quasi-ergodicity and related problems. On one hand, using an eigenfunction expansion for the transition density, we prove the existence and uniqueness of both quasi-stationary distribution (qsd) and mean ratio quasi-stationary distribution (mrqsd). The later is shown to be closely related to laws of large numbers (LLN) and to quasi-ergodicity. We further show that the mrqsd is the unique stationary distribution of a certain limiting ergodic diffusion process of the BM conditioned on not having been killed. We also show that a phase transition occurs from mrqsd to qsd. On the other hand, we study the large deviation behavior related to the above problems. A key observation is that the mrqsd is the unique minimum of certain large deviation rate function. We further prove that the limiting diffusion process also satisfies a large deviation principle with the rate function attaining its unique minimum at the mrqsd. These give interpretations of the mrqsd from different points of view, and establish some intrinsic connections among the above topics. Some general results concerning Yaglom limit, moment convergence and LLN are also obtained.     

Invariant tubular neighborhoods in infinite-dimensional Riemannian geometry,with applications to Yang-Mills theory

We present a new construction of tubular neighborhoods in (possibly infinite dimensional) Riemannian manifolds M, which allows us to show that if G is an arbitrary group acting isometrically on M, then every G-invariant submanifold with locally trivial normal bundle has a G-invariant total tubular neighborhood. We apply this result to the Morse strata of the Yang-Mills functional over a closed surface. The resulting neighborhoods play an important role in calculations of gauge-equivariant cohomology for moduli spaces of flat connections over non-orientable surfaces.     

The equations of elastostatics in a Riemannian manifold

To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify).     

A weak limit theorem for generalized multifractional Brownian motion

We provide a result on an approximation to the generalized multifractional Brownian motion in the space of continuous functions on [0, 1]. The construction of this approximation is based on the Poisson process.     

Every Sub-Riemannian Manifold Is the Gromov–Hausdorff Limit of a Sequence Riemannian Manifolds

In this paper, we will show that every sub-Riemannian manifold is the Gromov–Hausdorff limit of a sequence of Riemannian manifolds.     

Curve shortening in a Riemannian manifold

In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and , then for every n > 0, . Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S ¹, if γ₀ is a ramp, then we have a global flow which converges to a closed geodesic in C ^∞ norm. As an application, we prove the theorem of Lyusternik and Fet.      

Spherical-type hypersurfaces in a Riemannian manifold

We extend some rigidity results of Aleksandrov and Ros on compact hypersurfaces inR ⁿ to more general ambient spaces with the aid of the notion of almost conformal vector fields. These latter, at least locally, always exist and allow us to find interesting integral formulas fitting our purposes.     

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.     

An integral formula for the heat kernel of tubular neighborhoods of complete (connected) Riemannian manifolds

Let M be a complete connected Riemannian manifold and let N be a submanifold of M. Let v: E _v»N be the normal bundle of N and exp _v : E _v»M its exponential map.Let (exp _infv ^/sup-1 , M ₀) be the Fermi chart relative to the submanifold N. Then, by using the Fermi coordinates we obtain an integral formula for the Dirichlet heat kernel p _t ^m (-,-). That is, we obtain a probabilistic representation for the integral _N f(y)p _t ^M (x,y) dywhere f is any measurable function of compact support in M ₀. This representation involves a submanifold semi-classical Brownian Riemannian bridge process. Then applying the integral formula via a Riemannian submersion in [5], we obtain heat kernel formulae for the complex projective space cP ⁿ, the quaternionic projective space QP ⁿ and the Caley line CaP ¹. The case of the Caley plane CaP ² eludes us due to the lack of a submersion theorem.This work is part of a Ph.D. Thesis which was undertaken under Professor K. D. Elworthy, Mathematics Institute, Warwick University, Coventry CV47AL, England, Great Britain.     

The torsion of a submanifold of a Riemannian manifold

For a submanifoldM ⁿ of a Riemannian manifoldM ^q, the concept of a torsion bivector at the point x M ⁿ for given one- and two-dimensional directions fromT _x M ⁿ is introduced using only the first and second fundamental forms ofM ⁿ. Its relation to the concept of Gaussian torsion is then established. It is proved that: 1) equality to zero of the torsion bivector is necessary and, whenM ⁿ is a nondevelopable surface of a space of constant curvature with nonzero second fundamental form, is also sufficient for the "flattening" ofM ⁿ into some totally geodesicM ⁿ⁺¹ inM ^q; 2) when n = 2, the independence of the nonzero torsion bivector of direction characterizes a minimalM ² inM ^q.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 39–42, 1991.     

A central limit theorem for Gibbs measures relative to Brownian motion

We study a Gibbs measure over Brownian motion with a pair potential which depends only on the increments. Assuming a particular form of this pair potential, we establish that in the infinite volume limit the Gibbs measure can be viewed as Brownian motion moving in a dynamic random environment. Thereby we are in a position to use the technique of Kipnis and Varadhan and to prove a functional central limit theorem.     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.