De Rham cohomology of manifolds containing the points of infinite type,and Sobolev estimates for the\bar \partial - Neumann problem

We consider smooth bounded pseudoconvex domains Ω in Cⁿ whose boundary points of infinite type are contained in a smooth submanifoldM (with or without boundary) of the boundary having its (real) tangent space at each point contained in the null space of the Levi form ofbΩ at the point. (In particular, complex submanifolds satisfy this condition.) We consider a certain one-form α onbΩ and show that it represents a De Rham cohomology class on submanifolds of the kind described. We prove that if α represents the trivial cohomology class onM, then the Bergman projection and the \(\bar \partial - Neumann\) operator on Ω are continuous in Sobolev norms. This happens, in particular, ifM has trivial first De Rham cohomology, for instance, ifM is simply connected.     

Partial compactness for Landau-Lifshitz maxwell equation in two-dimension

We study the partial regularity of weak solutions to the 2-dimensional LandauLifshitz equations coupled with time dependent Maxwell equations by Ginzburg-Landau type approximation. Outside an energy concentration set of locally finite 2-dimensional parabolic Hausdorff measure, we prove the uniform local C ∞ bounds for the approaching solutions and then extract a subsequence converging to a global weak solution of the Landau-Lifshitz-Maxwell equations which are smooth away from finitely many points.     

On the regularity of a free boundary near contact points with a fixed boundary

We investigate the regularity of a free boundary near contact points with a fixed boundary, with C1,1 boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C¹ up to the fixed boundary. We will also construct some examples of irregular free boundaries.     

Totally geodesic submanifolds of the complex quadric

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Qm:=SO(m+2)/(SO(2)×SO(m)) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Qm by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Qm are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]: The first type is constituted by manifolds isometric to CP¹×RP¹; their existence follows from the fact that Q² is (via the Segre embedding) holomorphically isometric to CP¹×CP¹. The second type consists of 2-spheres of radius which are neither complex nor totally real in Qm.     

A notion of robustness and stability of manifolds

Starting from the notion of thickness of Parks we define a notion of robustness for arbitrary subsets of Rk and we investigate its relationship with the notion of positive reach of Federer. We prove that if a set M is robust, then its boundary ∂M is of positive reach and conversely (under very mild restrictions) if ∂M is of positive reach, then M is robust. We then prove that a closed non-empty robust set in Rk (different from Rk) is a codimension zero submanifold of class C¹ with boundary. As a partial converse we show that any compact codimension zero submanifold with boundary of class C² is robust. Using the notion of robustness we prove a kind of stability theorem for codimension zero compact submanifolds with boundary: two such submanifolds, whose boundaries are close enough (in the sense of Hausdorff distance), are diffeomorphic.     

On the generalized mean curvature

We study some properties of graphs whose mean curvature (in distributional sense) is a vector Radon measure. In particular, we prove that the distributional mean curvature of the graph of a Lipschitz continuous function u is a measure if and only if the distributional divergence of T u is a measure. This equivalence fails to be true if Lipschitz continuity is relaxed, as it is shown in a couple of examples. Finally, we prove a theorem of approximation in W ^(1,1) and in the sense of mean curvature of C ² graphs by polyhedral graphs. A number of examples illustrating different situations which can occur complete the work.     

Variational formulas of higher order mean curvatures

In this paper,we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional M2p of a submanifold M n in a general Riemannian manifold N n+m for p = 0,1,...,[n 2 ].As an example,we prove that closed complex submanifolds in complex projective spaces are critical points of the functional M2p,called relatively 2p-minimal submanifolds,for all p.At last,we discuss the relations between relatively 2p-minimal submanifolds and austere submanifolds in real space forms,as well as a special variational problem.     

On the piecewise smoothness of entropy solutions to scalar conservation laws

The behavior and structure of entropy solutions of scalar convex conservation laws are studied. It is well known that such entropy solutions consist of at most countable number of C¹-smooth regions. We obtain new upper. bounds on the higher order derivatives of the entropy solution in any one of its C¹-smoothness regions. These bounds enable us to measure the high order piecewise smoothness of the entropy solution. To this end we introduce an appropriate new Cⁿ-semi norm - localized to the smooth part of the entropy solution, and we show that the entropy solution is stable with respect to this norm. We also address the question regarding the number of C¹-smoothness pieces; we show that if the initial speed has a finite number of decreasing inflection points then it bounds the number of future shock discontinuities. Loosely speaking this says that in the case of such generic initial data the entropy solution consists of a finite number of smooth pieces, each of which is as smooth as the data permits. It is this type of piecewise smoothness which is assumed - sometime implicitly - in many finite-dimensional computations for such discontinuous problems.     

Compact quantum ergodic systems

We develop theory of multiplicity maps for compact quantum groups. As an application, we obtain a complete classification of right coideal C^∗-algebras of C(SUq(2)) for q∈[−1,1)?{0}. They are labeled with Dynkin diagrams, but classification results for positive and negative cases of q are different. Many of the coideals are quantum spheres or quotient spaces by quantum subgroups, but we do have other ones in our classification list.     

Conditional extremals in complete Riemannian manifolds

Conditional extremal curves in a complete Riemannian manifold M are defined as the critical points of the squared L² distance between the tangent vector field of a curve and a so-called prior vector field. We prove that this L² distance satisfies the Palais-Smale condition on the space of absolutely continuous curves joining two submanifolds of M, and thus establish the existence of critical points. We also prove a Morse index theorem in the case where the two submanifolds are single points, and use the Morse inequalities to place lower bounds on the number of critical points of each index.     

Absolute Points of Continuous and Smooth Polarities

This paper deals with sets of absolute points of continuous or smooth polarities in compact, connected or smooth projective planes, called topological polar unitals or smooth polar unitals, respectively. We will show that topological polar unitals are Z₂-homology spheres. In the four-dimensional case, a topological polar unital U is either a topological oval, or any line which intersects U in more than one point intersects in a set homeomorphic to S₁. Smooth polar unitals turn out to be smoothly embedded submanifolds of the point space. Moreover, secants intersect such unitals transversally. For these unitals, we will obtain full information on the existence of secants, tangents and exterior lines through given points according to their position with respect to the unital. The main result of this paper states that the possible dimensions of smooth polar unitals coincide with those of sets of absolute points of continuous polarities in the classical projective planes P₂F, F?{R,C,H,O}. Finally, we will prove that smooth polar unitals in four-dimensional smooth projective planes are topological ovals or are homeomorphic to S₃.     

SOME PROPERTIES OF HOLOMORPHIC CLIFFORDIAN FUNCTIONS IN COMPLEX CLIFFORD ANALYSIS

In this article, we mainly develop the foundation of a new function theory of several complex variables with values in a complex Clifford algebra defined on some subdomains of Cn+1, so-called complex holomorphic Cliffordian functions. We define the complex holomorphic Cliffordian functions, study polynomial and singular solutions of the equation D△mf=0, obtain the integral representation formula for the complex holomorphic Cliffordian functions with values in a complex Clifford algebra defined on some submanifolds of Cn+1, deduce the Taylor expansion and the Laurent expansion for them and prove an invariance under an action of Lie group for them.     

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.     

Smooth approximations

We prove, among other things, that a Lipschitz (or uniformly continuous) mapping f:X→Y can be approximated (even in a fine topology) by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a separable C(K) space (resp. super-reflexive space). Further, we show how smooth approximation of Lipschitz mappings is closely related to a smooth approximation of C¹-smooth mappings together with their first derivatives. As a corollary we obtain new results on smooth approximation of C¹-smooth mappings together with their first derivatives.     

Singular sets of planar hyperbolic billiards are regular

Many planar hyperbolic billiards are conjectured to be ergodic. This paper represents a first step towards the proof of this conjecture. The Hopf argument is a standard technique for proving the ergodicity of a smooth hyperbolic system. Under additional hypotheses, this technique also applies to certain hyperbolic systems with singularities, including hyperbolic billiards. The supplementary hypotheses concern the subset of the phase space where the system fails to be C ² differentiable. In this work, we give a detailed proof of one of these hypotheses for a large collection of planar hyperbolic billiards. Namely, we prove that the singular set and each of its iterations consist of a finite number of compact curves of class C ² with finitely many intersection points.     

Detectability of critical points of smooth functionals from their finite-dimensional approximations

Given a critical point of a C²-functional on a separable Hilbert space, we obtain sufficient conditions for it to be detectable (i.e. ‘visible’) from finite-dimensional Rayleigh-Ritz-Galerkin (RRG) approximations. While examples show that even nondegenerate critical points are, without any further restriction, not visible, we single out relevant classes of smooth functionals, e.g. the Hamiltonian action on the loop space or the functionals associated with boundary value problems for some semilinear elliptic equations, such that their nondegenerate critical points are visible from their RRG approximations.     

Second-order subdifferentials of C 1,1 functions and optimality conditions

We present second-order subdifferentials of Clarke's type of C ^1,1 functions, defined in Banach spaces with separable duals. One of them is an extension of the generalized Hessian matrix of such functions in ? ⁿ , considered by J. B. H.-Urruty, J. J. Strodiot and V. H. Nguyen. Various properties of these subdifferentials are proved. Second-order optimality conditions (necessary, sufficient) for constrained minimization problems with C ^1,1 data are obtained.     

Affine geometry of special real submanifolds of Cn

In this paper we propose an affine analogue and generalization of the geometry of special Lagrangian submanifolds of Cⁿ.      

Smooth functions in o-minimal structures

Fix an o-minimal expansion of the real exponential field that admits smooth cell decomposition. We study the density of definable smooth functions in the definable continuously differentiable functions with respect to the definable version of the Whitney topology. This implies that abstract definable smooth manifolds are affine. Moreover, abstract definable smooth manifolds are definably C^∞-diffeomorphic if and only if they are definably C¹-diffeomorphic.     

Area-preserving isotopies of self-transverse immersions of S 1 in ℝ2

Let C and C′ be two smooth self-transverse immersions of S ¹ into ?². Both C and C′ subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes C to C′ induces a one-to-one correspondence between the disks of C and C′. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking C to C′ is that there exists an isotopy for which the area of every disk of C equals that of the corresponding disk of C′. In this paper we show that this is also a sufficient condition.