arXiv research

Study inverse problems for twisted geodesic flows on manifolds.

Study proves uniqueness for ray transform on surfaces with obstacles.

Proves geodesic connections on 2-torus without invariant tori.

The paper shows how different geodesic flows on surfaces can be mapped to each other.

Characterizes parabolic flute surfaces with specific parameters.

Researchers prove twists can make surfaces parabolic even with large cuff lengths.

Study of twisted Ruelle zeta function on hyperbolic manifolds and its relation to analytic torsion.

Let $(X,ω)$ be a compact connected Kähler manifold and denote by $(\mathcal E^p,d_p)$ the metric completion of the space of Kähler potentials $\mathcal H_ω$ with respect to the $L^p$-type path length metric $d_p$. First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to $\mathcal E^p$ is …

We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,Ω) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the Hamiltonian flow of K has contractible periodic orbits with bounded period on all suffici…

We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equ…

We study the twisted Ruelle zeta function $ζ_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $ζ_X(0)$. In higher dimensions, we show more generally that $ζ_X(0)$ is locally constant with…

Study parabolicity of Riemann surfaces via Fenchel-Nielsen parameters.

The paper proves exponential mixing for hyperbolic manifolds, with applications to geodesic holonomy.

Study geodesic flow on symmetric surfaces to determine parabolic type.

Study of Veech surfaces and their twist tori on moduli spaces of abelian differentials.

The study finds infinitely many twist knot complements with totally geodesic surfaces.

The paper studies the twisted Calabi flow on Kähler manifolds.

We introduce a new object, the dynamical torsion, which extends the potentially ill-defined value at $0$ of the Ruelle zeta function of a contact Anosov flow twisted by an acyclic representation of the fundamental group. We show important properties of the dynamical torsion: it is invariant under deformations among con…

Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.

Paper studies convergence of Yang-Mills-Higgs flow on Kähler manifolds.

In this paper we present a certain class of geodesic vector fields of the double-twisted product R X R. Some examples of totally geodesic foliations are given.

The study examines Heegaard splittings defined by Dehn twists and finds hyperbolic metrics with specific geodesic lengths.

We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, we establish the existence of contractible closed characteristics on any thickening of the boundary of the neighborhood. When applied to…

For $φ$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $\int_X e^{-φ}.$ We prove that the logarithm of the volume is concave along continuous geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes f…

In this paper we study a generalization of the Kahler-Ricci flow, in which the Ricci form is twisted by a closed, non-negative (1,1)-form. We show that when a twisted Kahler-Einstein metric exists, then this twisted flow converges exponentially. This generalizes a result of Perelman on the convergence of the Kahler-Ric…

We introduce a pair of isospectral but non-isometric compact flat 3-manifolds called Tetra (a tetracosm) and Didi (a didicosm). The closed geodesics of Tetra and Didi are very different. Where Tetra has two quarter-twisting geodesics of the shortest length, Didi has four half-twisting geodesics. Nevertheless, these spa…

In this paper, we study the long-term behavior of the conical Kähler-Ricci flow on Fano manifold $M$. First, based on our work of locally uniform regularity for the twisted Kähler-Ricci flows, we obtain a long-time solution to the conical Kähler-Ricci flow by limiting a sequence of these twisted flows. Second, we study…

The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided th…

Study of gyroscopic Chaplygin systems and magnetic flows on spheres.

This paper studies generic properties of connections on vector bundles, solving cohomological equations and proving opaque connections.

Paper shows regularizing flow for conical Kähler-Ricci equations.

The paper proves uniqueness of closed timelike geodesics on de-Sitter tori with one singularity.

We prove a uniform isoperimetric inequality for all time along the twisted Kähler-Ricci flow on Fano manifolds.

We relate trimmed sums of twists in cylinders along a typical Teichmuller geodesic to the area Siegel-Veech constant.

Twisted $U$- and twisted $U/K$-hierarchies are soliton hierarchies introduced by Terng to find higher flows of the generalized sine-Gordon equation. Twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are among the most important classes of twisted hierarchies. In this paper, interesting first and higher flows of twi…

In the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we show that a geodesically complete hyperbolic surface is made up of its convex core with funnels attached along the simple closed geodesic components and half…

Study of J-flow on Kähler manifolds confirms energy properness.

We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle $(E,H_0)$ over a Riemann surface $X$. It is already known the gradient flow with initial data $(A_0,φ_0)$ converges to a critical point $(A_\infty, φ_\infty)$ of this functional. Using a modified Chern-Wei…

The paper connects geodesic flows, hyperbolic geodesics, and stable ergodicity.

Proves stability of geodesic flows on closed surfaces.

Geometrically, twist numbers on punctured tori are dense and non-continuous.

Homoclinic orbits found in geodesic flows on surfaces.

Study of curve shortening flow for twisted curves, defining curvature-torsion entropy.

Proves robust transitivity for geodesic flows from metrics with conjugate points.

Anosov geodesic flow proven in non-compact manifolds with negative curvature.

In this paper, by limiting twisted conical Kähler-Ricci flows, we prove the long-time existence and uniqueness of cusp Kähler-Ricci flow on compact Kähler manifold $M$ which carries a smooth hypersurface $D$ such that the twisted canonical bundle $K_M+D$ is ample. Furthermore, we prove that this flow converge to a uniq…

Study geodesic flows on cones over Riemannian manifolds, showing superintegrability.

Two Riemannian manifolds are said to have $C^k$-conjugate geodesic flows if there exist an $C^k$ diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step nilmanifold…