arXiv research

Study geodesics on ball quotients to find nonvanishing sections.

Study calculates global sections on complex curves.

The abstract proves that certain Reeb vector fields on 3-manifolds have Birkhoff sections.

The paper proves the existence of surfaces of section for geodesic flows on closed surfaces.

Constructs metrics with zero eigenvalues for Hodge-Laplacian.

Closed Riemannian manifolds with positive mixed sectional curvature

New results on geodesic flows using curve shortening flow.

Proves inequality for submanifolds with constant mean curvature.

Proves stability of geometric flows on compact manifolds.

We exhibit the first examples of closed 4-manifolds with nonnegative sectional curvature that lose this property when evolved via Ricci flow.

The paper extends trisection construction for Lefschetz fibrations with $(-n)$-sections.

Closed hyperbolic manifolds and manifolds with nonpositive sectional curvature are geometrically rigid under certain curvature conditions.

Survey on finding global surfaces of section for Reeb flows.

We prove that closed manifolds admitting a generic metric whose sectional curvature is locally quasi-constant are graphs of space forms. In the more general setting of QC spaces where sets of isotropic points are arbitrary, under suitable positivity assumption and for torsion-free fundamental groups they are still diff…

4-manifolds show every flat 3-manifold as cusp sections.

The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prov…

The study proves that certain 4-manifolds are Kähler if curvature is constant.

We show that in each dimension $4n+3$, $n\ge 1$, there exist infinite sequences of closed smooth simply connected manifolds $M$ of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these proper…

This paper deals with two aspects of relativistic cosmologies with closed (compact and boundless) spatial sections. These spacetimes are based on the theory of General Relativity, and admit a foliation into space sections S(t), which are spacelike hypersurfaces satisfying the postulate of the closure of space: each S(t…

Let D be a self-adjoint differential operator of Dirac type acting on sections in a vector bundle over a closed Riemannian manifold M. Let H be a closed D-invariant subspace of the Hilbert space of square integrable sections. Suppose D restricted to H is semibounded. We show that every element u in H has the weak uniqu…

Highly curved spaces have surfaces with many bumps.

We describe a construction of Riemannian metrics of nonnegative sectional curvature on a closed smooth nonorientable 4-manifold with fundamental group of order two that realizes a homotopy class that was not previously known to contain nonnegatively curved manifolds. The procedure yields new metrics of nonnegative sect…

After introducing the sub-Riemannian geometry of the Heisenberg group Hn, n \geq 1, we recall some basics about hypersurfaces endowed with the H-perimeter measure and horizontal Green's formulas. Then, we describe a class of compact closed hypersurfaces of constant horizontal mean curvature called "Isoperimetric Profil…

Extends Eells-Sampson theorem for manifolds with positive sectional curvature bounds.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

Proves a quantitative closing lemma for negatively curved manifolds.

This thesis predicts the distribution of smoothed zeros of random sections on line bundles.

Compactness theorem for Fueter sections yields non-zero harmonic 1-forms.

New algebraic characterization of sectional curvature bounds using Weitzenböck formulae.

Study embeds ruled surfaces into symplectic manifolds, finds Stein fillability results.

The paper characterizes Einstein 4-manifolds with semi-definite curvature and derives inequalities.

Entropy defined for submanifolds; applies to mean curvature flow limits of surfaces.

We introduce a geometric evolution equation for 3-manifolds with sectional curvature of one sign which is in some sense dual to the Ricci flow. On a closed 3-manifold with negative sectional curvature, we establish short time existence and a pair of monotonicity formulas for solutions to the flow. One of these formulas…

A bundle with base $B$ and fibre $F$ aspherical closed surfaces has a section if and only if the action $:π_1(B)\to{Out}(π_1(F))$ factors through $Aut(π_1(F))$ and a cohomology class is 0. We simplify and make more explicit the latter condition. We also show that the transgression $d^2_{2,0}$ in the homology LHS spectr…

The study finds sufficient conditions for Reeb flows to have genus zero global surfaces of section.

We find obstructions to the existence of Einstein metrics of non-negative sectional curvature on a smooth closed simply connected manifold of any dimension. The results are achieved by combining the classical Morse theory of the loop space with a new upper bound for the topological entropy of the geodesic flow in terms…

Paper shows how to transform certain flows into R-covered ones.

We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently Gromov-Hausdorff close to rays, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spect…

New invariant prevents minimal submanifolds in curved spaces.

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac14]$, and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This…

Study on 4D solitons with curvature constraints.

Let M be a connected Riemannian manifold and let D be a Dirac type operator acting on smooth compactly supported sections in a Hermitian vector bundle over M. Suppose D has a self-adjoint extension A in the Hilbert space of square-integrable sections. We show that any $L^2$-section $φ$ contained in a closed A-invariant…

The paper connects Bergman-Calabi diastasis to Kähler metrics with constant holomorphic sectional curvature.

We show that for a closed Riemannian manifold the quotient of the group of projective transformations by the group of isometries contains at most two elements unless the metric has constant positive sectional curvature or every projective transformation is an affine transformation.

We prove that compact Kähler manifolds whose sectional curvatures are close to 1/4-pinched have ratios of Chern numbers close to the corresponding ratios of a complex hyperbolic space form. We deduce that the Mostow-Siu surfaces (and their three-dimensional analogues constructed by the first author) do not admit Kähler…

The study explores metrics on real projective spaces with nonnegative sectional or positive Ricci curvature.

We show that all closed flat n-manifolds are diffeomorphic to a cusp cross-section in a finite volume hyperbolic (n+1)-orbifold.