arXiv research

Geodesic flow mixing on convex projective manifolds proven.

We establish exponential mixing for the geodesic flow $\varphi_t\colon T^1S\to T^1S$ of an incomplete, negatively curved surface $S$ with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil-Petersson flows for the moduli spaces ${\mathcal M}_{1,1}$ and ${\mathcal M}_{0,4}$ are expon…

Study equidistribution for flows on geometrically finite convergence group actions.

Develops a new method for learning discrete distributions without embedding them in a continuous space.

Study of Anosov flows using microlocal analysis for ergodicity and mixing properties.

Frame flows on certain symmetric spaces mix exponentially.

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

New proof for weak mixing in polygonal billiards.

Study proves projective Anosov subgroups lead to mixing flows in specific spaces.

This paper proves exponential mixing for frame flows on hyperbolic manifolds with cusps.

We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which are strictly convex by horospheres, we show the long time existence of mixed volu…

Study shows mixing of flows on specific geometric spaces.

This paper concerns the evolution of a closed hypersurface of dimension $n(\geq 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $β(\geq 1)$ of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, incl…

ContextFlow++ improves generative models by conditioning on mixed-variable contexts.

We introduce and study the flow of metrics on a foliated Riemannian manifold $(M,g)$, whose velocity along the orthogonal distribution is proportional to the mixed scalar curvature, $\Sc_{\,\rm mix}$. The flow is used to examine the question: When a foliation admits a metric with a given property of $\Sc_{\,\rm mix}$ (…

A flow of metrics, $g_t$, on a manifold is a solution of a differential equation $\dt g = S(g)$, where a geometric functional $S(g)$ is a symmetric $(0,2)$-tensor usually related to some kind of curvature. The mixed sectional curvature of a foliated manifold regulates the deviation of leaves along the leaf geodesics. W…

In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its e…

We apply conformal flows of metrics restricted to the orthogonal distribution $D$ of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive? Our evolution operator includes the integrability tensor of $D$, and for the case …

Let $Γ$ be a Zariski dense convex cocompact subgroup contained in an arithmetic lattice of $\operatorname{SO}(n, 1)^{\circ}$. We prove uniform exponential mixing of the geodesic flow for congruence covers of the hyperbolic manifold $Γ\backslash \mathbb{H}^n$ avoiding finitely many prime ideals. This extends the work of…

We study the dynamics of the Teichmuller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Holder observables. A geom…

We extend Teichmueller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of the action of the mapping class group of S on the deformation space. We observe…

Common perpendiculars equidistribute in negatively curved spaces.

Study on counting orbits and Poincaré series for specific hyperbolic metrics.

Ancient curve shortening flow in a disc with mixed boundary conditions is solved.

Study models Ricci flow on complex surfaces, showing mixed behavior.

A new method generates mixed-type features in tabular data with improved realism and accuracy.

MixFlows uses a mixture of flows for efficient variational inference.

Paper proposes a method to improve MCMC sampling for energy-based models.

If X is a proper CAT(-1)-space and $Γ$ a non-elementary discrete group of isometries acting properly discontinuously on X, it is shown that the geodesic flow on the quotient space Y=X/$Γ$ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary $\partial X$ and the non-wanderin…

The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.

We prove that a Ricci flow cannot develop a finite time singularity assuming the boundedness of a suitable space-time integral norm of the curvature tensor. Moreover, the extensibility of the flow is proved under a Ricci lower bound and the boundedness of a space-time integral norm of the scalar curvature.

New resonance theory for Anosov flows connects spectral properties to mixing measures.

We introduce Lagrangian mean curvature flow with boundary in Calabi--Yau manifolds by defining a natural mixed Dirichlet-Neumann boundary condition, and prove that under this flow, the Lagrangian condition is preserved. We also study in detail the flow of equivariant Lagrangian discs with boundary on the Lawlor neck an…

Flow on weighted graphs sharpens Bakry-Émery curvature.

PVI improves SIVI by directly optimizing ELBO without parametric assumptions.

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$, and $F_4/Spin(8)$. We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curv…

Geodesic flows on compact manifolds without conjugate points are shown to have a unique measure of maximal entropy.

Optimal self-distillation improves generative models' velocity risk and mode recovery.

New measure of maximal entropy found for a class of geometrically finite groups.

Injective flows for star-like manifolds improve variational inference efficiency.

Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen-Margulis' measure finiteness assumption used in recent work of Ricks is removed. We also construct examples of CAT(0) spaces which do not admit finite Bowen-Margulis measure.

Geodesic flows on certain surfaces are shown to be semi-conjugate to expansive flows.

Improves meta-learning efficiency with mixed-mode differentiation.

In this paper, we study the analytic continuation to complex time of the Hamiltonian flow of certain $G\times T$-invariant functions on the cotangent bundle of a compact connected Lie group $G$ with maximal torus $T$. Namely, we will take the Hamiltonian flows of one $G\times G$-invariant function, $h$, and one $G\time…

We prove the existence of C^{\infty} local solutions to a class of mixed type Monge-Ampere equations in the plane. More precisely, the equation changes type to finite order across two smooth curves intersecting transversely at a point. Existence of C^{\infty} global solutions to a corresponding class of linear mixed ty…

In this paper we investigate the flow of surfaces by a class of symmetric functions of the principal curvatures with a mixed volume constraint. We consider compact surfaces without boundary that can be written as a graph over a sphere. The linearisation of the resulting fully nonlinear PDE is used to prove a short time…

We prove that for closed surfaces $M$ with Riemannian metrics without conjugate points and genus $\geq 2$ the geodesic flow on the unit tangent bundle $T^1M$ has a unique measure of maximal entropy. Furthermore, this measure is fully supported on $T^1M$ and the flow is mixing with respect to this measure. We formulate …

The paper finds lower bounds for volumes of complex geometric structures.