Related papers: A smooth regularity theorem for nondegenerate CR-m…

Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-K\"ahler setting, we can then deduce a…

In this paper, we will study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $(u,v)$ from a smooth bounded open domain…

In this paper which is the first of a series of papers on smooth structures, the concepts of C-structures and smooth structures are introduced and studied. The notion of smooth structure on semi-integral domains is given. It is shown that…

We consider smooth deformations of the $CR$ structure of a smooth $2$-pseudoconcave compact $CR$ submanifold $\textsf{M}$ of a reduced complex analytic variety $\textsf{X}$ outside the intersection $D\,{\cap}\,\textsf{M}$ with the support…

Let M be a possibly noncompact manifold. We prove, generically in the C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This…

We consider normal compact surfaces $Y$ obtained from a minimal class VII surface $X$ by contraction of a cycle $C$ of $r$ rational curves with $C^2<0$. Our main result states that, if the obtained cusp is smoothable, then $Y$ is globally…

In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. After adapting the Almgren-Pitts min-max theory to…

We define cusp-decomposable manifolds and prove smooth rigidity within this class of manifolds. These manifolds generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume,…

Let X, Y be nonsingular real algebraic sets. A map fi:X-->Y is said to be k-regulous, where k is a nonnegative integer, if it is of class C^k and the restriction of fi to some Zariski open dense subset of X is a regular map. Assuming that Y…

We establish the existence and sharp global regularity results ($C^{0, \gamma}$, $C^{0, 1}$ and $C^{1, \alpha}$ estimates) for a class of fully nonlinear elliptic PDEs with unbalanced variable degeneracy. In a precise way, the degeneracy…

Let $K = R$ or $C$. We study basic invariants of submanifolds of solutions $\mathcal{M} = \{ y = Q(x,a,b)\} = \{b = P(a,x,y)\}$ in coordinates $x \in K^{n\geqslant 1}$, $y \in K$, $a \in K^{m\geqslant 1}$, $b \in K$ under…

We present a definable smooth version of the Thom transversality theorem. We show further that the set of non-transverse definable smooth maps is nowhere dense in the definable smooth topology. Finally, we prove a definable version of a…

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of…

We study smooth foliations of arbitrary codimension on homogeneous compact K\"ahler manifolds. We prove that smooth foliations on rational compact homogeneous manifolds are locally trivial fibrations and classify the smooth foliations with…

For Denjoy--Carleman differential function classes $C^M$ where the weight sequence $M=(M_k)$ is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is $C^M$ if it…

This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with…

Suppose that $N$ is a smooth manifold with a smooth Riemannian metric $g_0$, and that $\Gamma$ is a smooth submanifold of $N$. This paper proves that for a generic (in the sense of Baire category) smooth metric $g$ conformal to $g_0$, if…

We prove that in a class of non-equiregular sub-Riemannian manifolds corners are not length minimizing. This extends the results [4]. As an application of our main result we complete and simplify the analysis in [6], showing that in a…

The purpose of this paper has twofold. The first is to prove a unicity theorem for meromorphic mappings of a complete K\"{a}hler manifold M in P^n(C) sharing few hypersurfaces. The second is to give a unicity theorem for the case of…

It is a celebrated result of Mather that the group of $C^k$--diffeomorphisms of an $n$--manifold is simple, provided that a mild isotopy condition is satisfied, with the possible exception of $k=n+1$. The purpose of this article is mostly…