A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
New compactification of Teichmüller space via renormalized volume.
In this note, we prove a uniform distance distortion estimate for Ricci flows with uniformly bounded scalar curvature, independent of the lower bound of the initial -entropy. Our basic principle tells that once correctly renormalized, the metric-measure quantities obey similar estimates as in the non-collapsing case…
New adapted renormalized volume for hyperbolic 3-manifolds with compressible boundary.
Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the volume, or more precisely the ``dual volume'', of the convex core. On one hand, the…
We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifold is equal, up to an additive constant, to the volume of its convex core. We also provide a precise upper bound on the renormalized volume in terms of the Weil-Petersson distance between the conformal structures at infinity. As a consequence we…
In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.
Boundary distances determine conformal metrics
The universal Liouville action equals the renormalized volume of a hyperbolic 3-manifold.
The paper studies convex cocompact structures using Weil-Petersson flow.
Defines a new geometric quantity for hyperbolic manifolds, showing it's well-defined and invariant.
After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior und…
Score based learning (SBL) is a promising approach for learning Bayesian networks in the discrete domain. However, when employing SBL in the continuous domain, one is either forced to move the problem to the discrete domain or use metrics such as BIC/AIC, and these approaches are often lacking. Discretization can have …
Minimal surfaces in hyperbolic space have a renormalized area criterion.
Study uses renormalized area to determine metric expansion from minimal surfaces.
Defines renormalized volume for bounded regions in asymptotically hyperbolic Einstein spaces.
Study calculates the renormalized area of catenoids in hyperbolic spaces.
Defines and proves properties of weighted renormalized volume coefficients.
We study the critical points of the renormalized volume for acylindrical geometrically finite hyperbolic 3-manifolds that include rank-1 cusps, and show that the renormalized volume is locally convex around these critical points. We give a modified definition of the renormalized volume that is additive under gluing, an…
For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete Kähler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply the result to derive a renormalized Chern--Gauss--Bonnet formula. Relations between…
We study the renormalized volume of a conformally compact Einstein manifold. In even dimensions, we derive the analogue of the Chern-Gauss-Bonnet formula incorporating the renormalized volume. When the dimension is odd, we relate the renormalized volume to the conformal primitive of the -curvature. We show how all t…
We define and study the renormalized volume for geometrically finite hyperbolic -manifolds, including with rank- cusps. We prove a variation formula, and show that for certain families of convex co-compact hyperbolic metrics $g_\eps$ degenerating to a geometrically finite hyperbolic metric with rank- cus…
We interpret the physical -field renormalization group flow in the language of Courant algebroids, clarifying the sense in which this flow is the natural "Ricci flow" for generalized geometry. Next we show that the -field renormalization group flow preserves T-duality in a natural sense. As corollaries we obtain …
Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
Proves energy expression on Poincaré-Einstein spaces.
New properties are derived of renormalized volume functionals, which arise as coefficients in the asymptotic expansion of the volume of an asymptotically hyperbolic Einstein (AHE) manifold. A formula is given for the renormalized volume of an even-dimensional AHE manifold in terms of an arbitrary totally geodesic compa…
Renormalization in neural networks linked to quantum field theory.
We study the evolution of the renormalized volume functional for asymptotically Poincare-Einstein metrics (M,g) which are evolving by normalized Ricci flow. In particular, we prove that the time derivative of the renormalized volume along the flow is the negative integral of scal(g(t)) + n(n-1) over the manifold. This …
Renormalized pruning improves neural network accuracy.
Formula for renormalized area of hypersurfaces in hyperbolic spaces.
We compute renormalized curvature integrals on Poincaré-Einstein manifolds.
We study the infimum of the renormalized volume for convex-cocompact hyperbolic manifolds, as well as describing how a sequence converging to such values behaves. In particular, we show that the renormalized volume is continuous under the appropriate notion of limit. This result generalizes previous work in the subject…
New method calculates volume-renormalized mass from Hamiltonian perspective.
The paper studies circle packings using renormalization and subdivision rules.
Minimal surfaces in hyperbolic space have a sharp area bound.
We derive a new renormalized volume formula for conformally compact asymptotically hyperbolic manifolds in dimension four. The formula generalizes the ones given by Anderson, Albin, and Chang-Qing-Yang for the case of Poincare-Einstein manifolds. We also derive variational formulas for the renormalized seen as a functi…
The renormalized volume is reinterpreted using isoperimetric profiles.
A new Bayesian modeling method is proposed by combining the maximization of the marginal likelihood with a momentum-space renormalization group transformation for Gaussian graphical models. Moreover, we present a scheme for computint the statistical averages of hyperparameters and mean square errors in our proposed met…
The paper generalizes CR invariants using renormalized characteristic forms.
We introduce a natural definition of the renormalized volume of a 4-dimensional Ricci-flat ALE space. We then prove that the renormalized volume is always less or equal than zero, with equality if and only if the ALE space is isometric to its asymptotic cone. Currently the only known examples of 4-dimensional Ricci-fla…
The Gauss-Bonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the -cohomology spaces as well as to carry out the heat equa…
We propose a new statistical model suitable for machine learning of systems with long distance correlations such as natural languages. The model is based on directed acyclic graph decorated by multi-linear tensor maps in the vertices and vector spaces in the edges, called tensor network. Such tensor networks have been …
A method to analyze maps into circles with singularities.
We survey the renormalized volume of hyperbolic 3-manifolds, as a tool for Teichmuller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen's quasifuchsian (or more generally Kleinian) reciprocity, for which different arguments…
Renormalized volume invariant for knots in 3-sphere computed.
The paper constructs global CR invariants from renormalized characteristic forms.
We prove that the renormalized volume of almost-Fuchsian hyperbolic -manifolds is non-negative, with equality only for Fuchsian manifolds.
We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic of a four dimensional manifold-with-boundary in terms of the geometry of the Loewner-Nirenberg singular Yamabe metric in a prescribed conformal class. The formula involves the renormalized volume and a boundary integral. It is shown that if the…
We discuss from a geometric point of view the connection between the renormalization group flow for non--linear sigma models and the Ricci flow. This offers new perspectives in providing a geometrical landscape for 2D quantum field theories. In particular we argue that the structure of Ricci flow singularities suggests…