Bounds for the first eigenvalue of the horizontal Laplacian in positively curved sub-Riemannian manifolds

We establish lower bounds for the first non-zero eigenvalue for the natural geometric sub-elliptic Laplacian operator defined on sub-Riemannian manifolds of step 2 that satisfy a positive curvature condition. The methods are very general and can be applied even when the sub-Riemannian geometry has considerable torsion.     

A topological splitting theorem for sub-Riemannian manifolds

We prove an analogue of the Cheeger–Gromoll splitting theorem for sub-Riemannian manifolds with the measure contraction property instead of the nonnegativity of the Ricci curvature. If such a sub-Riemannian manifold contains a straight line, then the manifold splits diffeomorphically, where the splitting is not necessarily isometric. We prove that such a sub-Riemannian manifold containing a straight line cannot split isometrically under some typical condition in sub-Riemannian geometry. Heisenberg groups are such examples.     

Volume and distance comparison theorems for sub-Riemannian manifolds

In this paper we study global distance estimates and uniform local volume estimates in a large class of sub-Riemannian manifolds. Our main device is the generalized curvature dimension inequality introduced by the first and the third author in [3] and its use to obtain sharp inequalities for solutions of the sub-Riemannian heat equation. As a consequence, we obtain a Gromov type precompactness theorem for the class of sub-Riemannian manifolds whose generalized Ricci curvature is bounded from below in the sense of [3].     

On the functional form of convex underestimators for twice continuously differentiable functions

The optimal functional form of convex underestimators for general twice continuously differentiable functions is of major importance in deterministic global optimization. In this paper, we provide new theoretical results that address the classes of optimal functional forms for the convex underestimators. These are derived based on the properties of shift-invariance and sign- invariance.     

On the canonical extension of a differentiable manifold

In this paper, the differential geometry of second canonical extension² M of a differentiable manifoldM is studied. Some vector fields tangent to² M inTTM are determined. In addition we obtain that the second canonical extensions ofM and a totally geodesic submanifold inM are totally geodesic submanifolds inTTM and² M respectively.     

The abelian monodromy extension property for families of curves

Necessary and sufficient conditions are given (in terms of monodromy) for extending a family of smooth curves over an open subset to a family of stable curves over S. More precisely, we introduce the abelian monodromy extension (AME) property and show that the standard Deligne–Mumford compactification is the unique, maximal AME compactification of the moduli space of curves. We also show that the Baily–Borel compactification is the unique, maximal projective AME compactification of the moduli space of abelian varieties.     

Locally defined operators in the space of Whitney differentiable functions

For a given set A⊂Rⁿ$A\subset {\mathbb{R}}^n$ a representation theorem for locally defined operators mapping the space C^m(A)$C^m\left(A\right)$ of Whitney differentiable functions into C⁰(A)$C^0\left(A\right)$ is presented.     

Stochastic completeness and volume growth in sub-Riemannian manifolds

In this note we generalize A. Grigor’yan’s volume test for the stochastic completeness of a Riemannian manifold to a sub-Riemannian setting. As an application of this result, and of a new estimate of the growth of the volume of the metric balls at infinity, we give a different proof of (and extend) a theorem in Baudoin and Garofalo (Arxiv preprint, submitted paper, 2009) stating that when a smooth, complete and connected manifold satisfies the generalized curvature-dimension inequality introduced in that paper, then the manifold turns out to be stochastically complete.     

On the uniqueness of the coincidence index on orientable differentiable manifolds

The fixed point index of topological fixed point theory is a well studied integer-valued algebraic invariant of a mapping which can be characterized by a small set of axioms. The coincidence index is an extension of the concept to topological (Nielsen) coincidence theory. We demonstrate that three natural axioms are sufficient to characterize the coincidence index in the setting of continuous mappings on oriented differentiable manifolds, the most common setting for Nielsen coincidence theory.     

On genericity for holomorphic curves in four-dimensional almost-complex manifolds

We consider spaces of immersed (pseudo-)holomorphic curves in an almost complex manifold of dimension four. We assume that they are either closed or compact with boundary in a fixed totally real surface, so that the equation for these curves is elliptic and has a Fredholm index. We prove that this equation is regular if the Chern class is ≥ 1 (in the case with boundary, if the ambient Maslov number is ≥ 1). Then the spaces of holomorphic curves considered will be manifolds of dimension equal to the index.     

Representation theorem for locally defined operators in the space of Whitney differentiable functions

Under the assumption that is perfect, a representation theorem for locally defined operators mapping the space C ^m (A) of Whitney differentiable functions into C ¹(A) is given and an open problem is presented.     

On the property of kelley in the hyperspace and Whitney continua

In this paper, we introduce the notion of property [K]¹ which implies property [K], and we show the following: Let X be a continuum and let ω be any Whitney map for C(X). Then the following are equivalent. (1) X has property [K]¹. (2) C(X) has property [K]¹. (3) The Whitney continuum ω⁻¹(t) (0⩽t<ω(X)) has property [K]¹.As a corollary, we obtain that if a continuum X has property [K]¹, then C(X) has property [K] and each Whitney continuum in C(X) has property [K]. These are partial answers to Nadler's question and Wardle's question ([10, (16.37)] and [11, p. 295]).Also, we show that if each continuum X_n (n=1,2,3,…) has property [K]¹, then the product ∏X_n has property [K]¹, hence C(∏X_n) and each Whitney continuum have property [K]¹. It is known that there exists a curve X such that X has property [K], but X×X does not have property [K] (see [11]).     

On the congruence extension property

It is shown that a quasivariety has the extension property for all relative congruences if it has the extension property for principal relative congruences. This generalizes the analogous result for varieties, due to A. Day. The proof differs from Day's in that it avoids the use of Zorn's Lemma. Received February 13, 1997; accepted in final form January 19, 1998.     

On the role of continuously differentiable exact penalty functions in constrained global optimization

The aim of this paper is to show that the new continuously differentiable exact penalty functions recently proposed in literature can play an important role in the field of constrained global optimization. In fact they allow us to transfer ideas and results proposed in unconstrained global optimization to the constrained case.First, by drawing our inspiration from the unconstrained case and by using the strong exactness properties of a particular continuously differentiable penalty function, we propose a sufficient condition for a local constrained minimum point to be global.Then we show that every constrained local minimum point satisfying the second order sufficient conditions is an attraction point for a particular implementable minimization algorithm based on the considered penalty function. This result can be used to define new classes of global algorithms for the solution of general constrained global minimization problems. As an example, in this paper we describe a simulated annealing algorithm which produces a sequence of points converging in probability to a global minimum of the original constrained problem.     

First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds

We calculate the first and second variation formulae for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that move the singular set of a ${\mathcal{C}^2}$ surface and non-singular variations for ${\mathcal{C}^2_\mathcal{H}}$ surfaces. These formulae enable us to construct a stability operator for non-singular ${\mathcal{C}^2}$ surfaces and another one for ${\mathcal{C}^2}$ (eventually singular) surfaces. Then we can obtain a necessary condition for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in terms of the pseudo-hermitian torsion and the Webster scalar curvature. Finally we give a classification of the complete stable surfaces in the roto-translation group ${\mathcal{R}\mathcal{T}}$ .     

Matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions

The author examines matrices and bordered matrices having exactly one or at most one negative eigenvalue. These results are then used to state matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions. For quadratic functions, a new criterion for quasiconvexity is established, and its equivalence to the already known criterion [11,23] is also shown.