arXiv research

Study of flows on complex manifolds with holomorphic properties.

Study of flows on 7D manifolds with holomorphic properties.

We show that Perelman's W-functional can be generalized to Sasaki-Ricci flow. When the basic first Chern class is positive, we prove a uniform bound on the scalar curvature, the diameter and a uniform $C^1$ bound for the transverse Ricci potential along the Sasaki-Ricci flow, which generalizes Perelman's results Kahler…

The study connects taut contact circles to transversely holomorphic flows on spherical 3-manifolds.

Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as tra…

Study on stability of Sasaki structures under deformations.

Given a Sasaki manifold S, we prove the Sasaki-Ricci flow converges exponentially fast to a Sasaki-Einstein metric if one exists, provided the automorphism group of the transverse holomorphic structure is trivial.

Formula calculates residues for maps near holomorphic distributions.

We introduce a holomorphic sheaf E on a Sasaki manifold and study two new notions of stability for E along the Sasaki-Ricci flow related to the `jumping up' of the number of global holomorphic sections of E at infinity. First, we show that if the Mabuchi K-energy is bounded below, the transverse Riemann tensor is bound…

Study on complex tori foliations and flat geometries.

We study the space of Sasaki metrics on a compact manifold $M$ by introducing an odd-dimensional analogue of the $J$-flow. That leads to the notion of critical metric in the Sasakian context. In analogy to the Kähler case, on a polarised Sasakian manifold there exists at most one normalised critical metric. The flow is…

Topology of Foliations of the Riemann Surfaces given by the real part of generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB) instead of using just one closed transversal curve as in the classical approach of the ergodic theory. In some cases the TC…

Holomorphic branched Cartan geometry defined on complex manifolds.

The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.

Establishes jet transversality for regular maps from flexible manifolds.

Study transverse $J$-holomorphic curves linking nearly Kähler $\mathbb{CP}^3$ to minimal surfaces.

Adapts Frolicher-type inequalities to foliations.

We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally to the study of the case of foliations with singularities. A first extension theo…

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 zero measure subset of the manifold.

The paper establishes Schwarz type lemmas for pseudo-Hermitian manifolds.

Topology of the Generic Hamiltonian Dynamical Systems on the Riemann Surfaces given by the real part of the generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB). This approach allows us to present a convenient combinatorial model of the whole topolo…

In this paper, we consider holomorphic mappings between real hypersurfaces in different dimensional complex spaces. We give a number of conditions that imply that such mappings are transversal to the target hypersurface at most points.

Extends symplectic flow results to foliations.

New systems derived from Hilbert schemes on surfaces.

Study blow-ups in generalized complex geometry using holomorphic ideals.

Hodge numbers of Sasakian manifolds remain unchanged under deformations.

Let F be a Kähler foliation on a compact Riemannian manifold M. we study the properties of infinitesimal automorphisms on (M,F), and in particular we concentrate on the transversal conformal field, transversal projective field and transversally holomorphic field

In this paper an analytic proof of a generalization of a theorem of Bismut ([Bis1, Theorem 5.1]) is given, which says that, when $v$ is a transversal holomorphic vector field on a compact complex manifold $X$ with a zero point set $Y$, the embedding $j:Y\to X$ induces a natural isomorphism between the holomorphic equiv…

Paper introduces entropy for Riemannian foliations and connects it to Ricci flow.

In this paper we study (smooth and holomorphic) foliations which are invariant under transverse actions of Lie groups.

Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of it…

We introduce transverse Chern-Ricci flow for transversely Hermitian foliations, which is analogous to the Chern-Ricci flow. We show that when $\mathcal{F}$ is homologically orientable and the basic first Bott-Chern class is zero, starting at any transversely Hermitian metric the flow exists for all time and as $t\right…

We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and …

No projective structure found on foliations of elliptic curves.

Survey on holomorphic structures on complex manifolds.

Constructs the moduli space of super J-holomorphic curves.

Reeb flow made transverse to foliations without invariant measures.

We study the classification of singularities of holomorphic foliations and non-integrable one-forms under the hypothesis of transversality with real hypersurfaces.

The study introduces new foliations and structures on complex manifolds.

Study on transverse Ricci solitons on compact foliated manifolds.

Study deformations of Calabi-Yau foliations using Kuranishi spaces.

We introduce two new functionals on Sasaki manifolds, inspired by the work of Perelman, which are monotonic along the Sasaki-Ricci flow. We relate their gradient flow, via diffeomorphisms preserving the foliated structure of the manifold, to the transverse Ricci flow. Finally, when the basic first Chern class is positi…

The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.

Characterizes transverse surfaces for pseudo-Anosov flows in 3-manifolds.

We prove a general result about the short time existence and uniqueness of second order geometric flows transverse to a Riemannian foliation on a compact manifold. Our result includes some flows already existing in literature, as the transverse Ricci flow, the Sasaki-Ricci flow and the Sasaki J-flow and motivates the s…

Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.

We classify simply connected compact Sasaki manifolds of dimension $2n+1$ with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be bi-holomorphic to $\C^{n+1}\backslash \{0\}$. As an application we recover the Mori-Siu-Yau theorem on the Frankel conjecture a…

We prove the existence of foliations transverse to pseudo-Anosov flows using veering triangulations.