Relative isoperimetric inequality and¶linear isoperimetric inequality for minimal submanifolds

We prove an optimal relative isoperimetric inequality

Second variation and Legendrian stabilities of minimal Legendrian submanifolds in Sasakian manifolds

First, we derive a new second variation formula which holds for minimal Legendrian submanifolds in Sasakian manifolds. Using this, we prove that any minimal Legendrian submanifold in an η-Einstein Sasakian manifold with “nonpositive” η-Ricci constant is stable. Next we introduce the notion of the Legendrian stability of minimal Legendrian submanifolds in Sasakian manifolds. Using our second variation formula, we find a general criterion for the Legendrian stability of minimal Legendrian submanifolds in η-Einstein Sasakian manifolds with “positive” η-Ricci constant.     

New monotonicity formulas for Ricci curvature and applications. I

We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov?CHausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop?CGromov volume comparison theorem and Perelman??s celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these. Moreover, we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the Green function that we prove. This is parallel to the fact that Perelman??s monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li?CYau. In [CM4] one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross-sections of Einstein manifolds. Finally, there are obvious parallelisms between our monotonicity and the positive mass theorem of Schoen?CYau and Witten.     

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.     

Estimate and monotonicity of the first eigenvalue under the Ricci flow

In this paper, we first derive a monotonicity formula for the first eigenvalue of on a closed surface with nonnegative scalar curvature under the (unnormalized) Ricci flow. We then derive a general evolution formula for the first eigenvalue under the normalized Ricci flow. As an application, we obtain various monotonicity formulae and estimates for the first eigenvalue on closed surfaces.     

On compact splitting complex submanifolds of quotients of bounded symmetric domains

We study compact complex submanifolds S of quotient manifolds X = ?/Γ of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms, and we are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifolds S ? X, i.e., under the assumption that the tangent sequence over S splits holomorphically. We prove results of two types. The first type of results concerns S ? X which are characteristic complex submanifolds, i.e., embedding ? as an open subset of its compact dual manifold M by means of the Borel embedding, the non-zero(1, 0)-vectors tangent to S lift under a local inverse of the universal covering map π : ? → X to minimal rational tangents of M.We prove that a compact characteristic complex submanifold S ? X is necessarily totally geodesic whenever S is a splitting complex submanifold. Our proof generalizes the case of the characterization of totally geodesic complex submanifolds of quotients of the complex unit ball Bnobtained by Mok(2005). The proof given here is however new and it is based on a monotonic property of curvatures of Hermitian holomorphic vector subbundles of Hermitian holomorphic vector bundles and on exploiting the splitting of the tangent sequence to identify the holomorphic tangent bundle TSas a quotient bundle rather than as a subbundle of the restriction of the holomorphic tangent bundle TXto S. The second type of results concerns characterization of total geodesic submanifolds among compact splitting complex submanifolds S ? X deduced from the results of Aubin(1978)and Yau(1978) which imply the existence of K¨ahler-Einstein metrics on S ? X. We prove that compact splitting complex submanifolds S ? X of sufficiently large dimension(depending on ?) are necessarily totally geodesic. The proof relies on the Hermitian-Einstein property of holomorphic vector bundles associated to TS,which implies that endomorphisms of such bundles are parallel, and the construction of endomorphisms of these vector bundles by means of the splitting of the tangent sequence on S. We conclude with conjectures on the sharp lower bound on dim(S) guaranteeing total geodesy of S ? X for the case of the type-I domains of rank2 and the case of type-IV domains, and examine a case which is critical for both conjectures, i.e., on compact complex surfaces of quotients of the 4-dimensional Lie ball, equivalently the 4-dimensional type-I domain dual to the Grassmannian of 2-planes in C~4.     

Sharp and simple bounds for the Erlang delay and loss formulae

We prove some simple and sharp lower and upper bounds for the Erlang delay and loss formulae and for the number of servers that invert the Erlang delay and loss formulae. We also suggest simple and sharp approximations for the number of servers that invert the Erlang delay and loss formulae. We illustrate the importance of these bounds by using them to establish convexity proofs. We show that the probability that the M/M/s queue is empty is a decreasing and convex function of the traffic intensity. We also give a very short proof to show that the Erlang delay formula is convex in the traffic intensity when the number of servers is held constant. The complete proof of this classical result has never been published. We also give a very short proof to show that the Erlang delay formula is a convex function of the (positive integer) number of servers. One of our results is then used to get a sharp bound to the Flow Assignment Problem.     

Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension

For compact Riemannian manifolds with convex boundary, B. White proved the following alternative: either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B. White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four–dimensional ball B ⁴ with the following properties: (i) B ⁴ has strictly convex boundary. (ii) There exists a complete nonconstant geodesic ${c : \mathbb{R} \to B^4}$ . (iii) There does not exist a closed geodesic in B ⁴.     

An isoperimetric inequality related to a Bernoulli problem

Given a bounded domain Ω we look at the minimal parameter Λ(Ω) for which a Bernoulli free boundary value problem for the p-Laplacian has a solution minimising an energy functional. We show that amongst all domains of equal volume Λ(Ω) is minimal for the ball. Moreover, we show that the inequality is sharp with essentially only the ball minimising Λ(Ω). This resolves a problem related to a question asked in Flucher et al. (Reine Angew Math 486:165–204, 1997).     

A minimizing deformation of Legendrian submanifolds in the standard sphere

In this note we study the moduli space of minimal Legendrian submanifolds in the standard sphere S²ⁿ⁻¹. We show that new examples of minimal Legendrian submanifolds can be constructed, if we can solve a certain equation for a function on a nearby glued Legendrian submanifold. As a step toward solving this equation, we prove short-time existence for a particular gradient flow on the space of immersed Legendrian submanifolds. A new necessary condition for a Lagrangian embedding into is given.     

Higher order optimal approximation of Csiszar's f-divergence

Here are established various sharp and nearly optimal probabilistic inequalities giving the high order approximation of Csiszar's f-divergence between two probability measures, which is the most essential and general tool for their comparison. The above are done through Taylor's formula, generalized Taylor–Widder's formula, an alternative recent expansion formula. Based on these we give many representation formulae of Csiszar's distance, then we estimate in all directions their remainders by using either the norms approach or the modulus of continuity way. Most of the last probabilistic estimates are sharp or nearly sharp, attained by basic simple functions.     

A new expression for the density of totally geodesic submanifolds in space forms, with stereological applications

Integral section formulae for totally geodesic submanifolds (planes) intersecting a compact submanifold in a space form are available from appropriate representations of the motion invariant density (measure) of these planes. Here we present a new decomposition of the invariant density of planes in space forms. We apply the new decomposition to rewrite Santaló's sectioning formula and thereby to obtain new mean values for lines meeting a convex body. In particular we extend to space forms a recently published stereological formula valid for isotropic plane sections through a fixed point of a convex body in R³.     

Eigenvalues of the Laplacian and extrinsic geometry

We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second, we consider complex submanifolds of the complex projective space $\mathbb C P^N$ instead of submanifolds of $\mathbb R ^N$ and we obtain an eigenvalue upper bound depending only on the dimension of the submanifold which is sharp for the first non-zero eigenvalue.     

No Dimension-Independent Core-Sets for Containment Under Homothetics

This paper deals with the containment problem under homothetics which has the minimal enclosing ball (MEB) problem as a prominent representative. We connect the problem to results in classic convex geometry and introduce a new series of radii, which we call core-radii. For the MEB problem, these radii have already been considered from a different point of view and sharp inequalities between them are known. In this paper sharp inequalities between core-radii for general containment under homothetics are obtained. Moreover, the presented inequalities are used to derive sharp upper bounds on the size of core-sets for containment under homothetics. In the MEB case, this yields a tight (dimension-independent) bound for the size of such core-sets. In the general case, we show that there are core-sets of size linear in the dimension and that this bound stays sharp even if the container is required to be symmetric.     

Sub-harmonicity,monotonicity formula and finite Morse index solutions of an elliptic equation with negative exponent

A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with "negative exponent" is established.It is well known that such a monotonicity formula plays an essential role in the study of finite Morse index solutions of equations with "positive exponent".Unlike the positive exponent case,we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.     

Extrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motions in the extrinsic balls, i.e. lower bounds for the time it takes (on average) for Brownian particles to diffuse within the extrinsic ball from a given starting point before they hit the boundary of the extrinsic ball. In those cases, where we may extend our analysis to hold all the way to infinity, we apply a capacity comparison technique to obtain a sufficient condition for the submanifolds to be parabolic, i.e. a condition which will guarantee that any Brownian particle, which is free to move around in the whole submanifold, is bound to eventually revisit any given neighborhood of its starting point with probability 1. The results of this paper are in a rough sense dual to similar results obtained previously by the present authors in complementary settings where we assume that the curvatures are bounded from above.     

A Coarea Formula for Smooth Contact Mappings of Carnot–Carathéodory Spaces

We prove the coarea formula for sufficiently smooth contact mappings of Carnot manifolds to Carnot–Carathéodory spaces. In particular, we investigate level surfaces of these mappings, and compare Riemannian and sub-Riemannian measures on them. Our main tool is the sharp asymptotic behavior of the Riemannian measure of the intersection of a tangent plane to a level surface and a sub-Riemannian ball. This calculation in particular implies that the sub-Riemannian measure of the set of characteristic points (i.e., the points at which the sub-Riemannian differential is degenerate) equals zero on almost every level set.     

Regular submanifolds, graphs and area formula in Heisenberg groups

We describe intrinsically regular submanifolds in Heisenberg groups Hn. Low dimensional and low codimensional submanifolds turn out to be of a very different nature. The first ones are Legendrian surfaces, while low codimensional ones are more general objects, possibly non-Euclidean rectifiable. Nevertheless we prove that they are graphs in a natural group way, as well as that an area formula holds for the intrinsic Hausdorff measure. Finally, they can be seen as Federer-Fleming currents given a natural complex of differential forms on Hn.     

Sharp upper bounds on the number of the scattering poles

We study the scattering poles of a compactly supported “black box” perturbations of the Laplacian in Rⁿ, n odd. We prove a sharp upper bound of the counting function N(r) modulo o(rⁿ) in terms of the counting function of the reference operator in the smallest ball around the black box. In the most interesting cases, we prove a bound of the type N(r)?A_nrⁿ+o(rⁿ) with an explicit A_n. We prove that this bound is sharp in a few special spherically symmetric cases where the bound turns into an asymptotic formula.