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The existence of closed hypersurfaces of prescribed curvature in semi-riemannian manifolds is proved provided there are barriers.
We show that complex symplectic structures need not be preserved under small deformations, and we find sufficient conditions for this to happen. We study various cohomologies of compact complex symplectic manifolds, obtaining some…
In this paper we study the positivity of the cotangent bundle of projective manifolds. We conjecture that the cotangent bundle is pseudoeffective if and only the manifold has non-zero symmetric differentials. We confirm this conjecture for…
We prove that there exist infinitely many topologically slice knots which cannot bound a smooth null-homologous disk in any definite 4-manifold. Furthermore, we show that we can take such knots so that they are linearly independent in the…
We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.
A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable leaf if and only if the set of compact leaves is not a meager subset of the manifold.
In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to…
Certain families of manifolds which support Anosov flows do not support expanding, quasi-isometric foliations.
We prove that any weakly symplectically fillable contact manifold is tight. Furthermore we verify the strong Weinstein conjecture for contact manifolds that appear as the concave boundary of a directed symplectic cobordism whose positive…
We formulate a conjecture that arithmetic locally symmetric manifolds have simple homotopy type, and prove it for the non-compact case. More precisely, we show that, for any symmetric space S of non-compact type without Euclidean de Rham…
We construct holomorphic maps with a Siegel disk whose boundary is not locally connected (and is an indecomposable continuum), yet compactly contained in the domain of definition of the map. Our examples are injective and defined on a…
In recent years the conformal bootstrap has produced surprisingly tight bounds on many non-perturbative CFTs. It is an open question whether such bounds are indeed saturated by these CFTs. A toy version of this question appears in a recent…
Some properties of non-orientable 3-manifolds are shown. The semi-group of cobordism of immersions of surfaces in such manifolds is computed and proven actually to be a group. Explicit invariants are provided.
In this paper we deal with Seifert fibre spaces, which are compact 3-manifolds admitting a foliation by circles. We give a combinatorial description for these manifolds in all the possible cases: orientable, non-orientable, closed, with…
We provide examples of classical small-cancellation groups which have non-sigma-compact Morse boundary. These are first known examples of groups with non-sigma-compact Morse boundary. Some small-cancellation groups do have sigma-compact…
We study the possibility of realizing exotic smooth structures on punctured simply connected $4$-manifolds as leaves of a codimension one foliation on a compact manifold. In particular, we show the existence of uncountably many smooth open…
We study nonnegative solutions to the Fractional Porous Medium Equation on a suitable class of connected, noncompact Riemannian manifolds. We provide existence and smoothing estimates for solutions, in an appropriate weak (dual) sense, for…
We follow our study of non-K\"ahler complex structures on $R^4$ that we defined in a previous paper. We prove that these complex surfaces do not admit any smooth complex compactification. Moreover, we give an explicit description of their…
We show that the hyperbolic manifold $\mathbb{H}^n/\mathbb{Z}^{n-2}$ is not rigid under all compactly supported deformations that preserve the scalar curvature lower bound $-n(n-1)$, and that it is rigid under deformations that are further…
This article is devoted to the geometric construction which states a natural correspondence between topological coverings of a foliated manifolds and noncommutative coverings of the operator algebras. However this correspondence is not one…