Minimal surfaces in hyperbolic space have a renormalized area criterion.
problem Minimal surfaces in hyperbolic space
method Renormalized area criterion
result Y must be a totally geodesic disk
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48 results for “Renormalized Area”
Study calculates the renormalized area of catenoids in hyperbolic spaces.
problem Calculating the renormalized area of catenoids in hyperbolic spaces.
method Variational characterization and Chern--Gauss--Bonnet formulas for locally conformally flat manifolds.
result Renormalized area of catenoids varies continuously from negative infinity to twice the area of totally geodesic hypersurfaces.
Study uses renormalized area to determine metric expansion from minimal surfaces.
problem Recovering the expansion of asymptotically hyperbolic metrics from minimal surfaces.
method Uses renormalized area functional on minimal submanifolds to recover metric expansion.
result Proves rigidity for log-analytic metrics and determines obstruction tensor.
Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
problem Calculating the renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
method Derives a Gauss-Bonnet formula for renormalized area in terms of scalar invariants and characterizes minimal hypersurfaces in terms of conformal geometry.
result Derives a formula expressing renormalized area in terms of integrals of scalar Riemannian invariants.
A Sharp Isoperimetric Property of the Renormalized Area of a Minimal Surface in Hyperbolic Spacemath.DG
Minimal surfaces in hyperbolic space have a sharp area bound.
problem Bounding the renormalized area of minimal surfaces.
method Proving an inequality using conformal length of ideal boundary.
result Sharp isoperimetric property of renormalized area.
Formula for renormalized area of hypersurfaces in hyperbolic spaces.
problem Calculating the renormalized area of asymptotically minimal hypersurfaces in hyperbolic spaces.
method Combining Chen's conformal invariant quantity and Chern-Gauss-Bonnet formulas.
result Extension of renormalized area formulas to higher dimensions and non-minimal cases.
Neumann Data and Second Variation Formula of Renormalized Area for Conformally Compact Static Spacesmath.DG
Paper derives formulas for static Einstein spaces, linking Neumann data to stability.
problem Stability of conformally compact static spaces.
method First and second variation formulas for renormalized area.
result Negativity of Neumann data implies instability.
Formula derived for renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
problem Calculating the renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
method Decomposition of extrinsic -curvature and application to renormalized area.
result Renormalized area formula expressed as a linear combination of Euler characteristic and scalar conformal submanifold invariant.
If is a properly embedded minimal surface in a convex cocompact hyperbolic 3-manifold with boundary at infinity an embedded curve , then Graham and Witten showed how to define a renormalized area $\calA$ of via Hadamard regularization. We study renormalized area as a functional on the space of all such m…
Solves Dirichlet problem for harmonic maps to give geodesic insights.
problem Asymptotic Dirichlet problem for harmonic maps.
method Holographic characterization using conformal geodesics.
result Characterizes conformal geodesics on the boundary.
This article describes some geometric invariants and conformal anomalies for conformally compact Einstein manifolds and their minimal submanifolds which have recently been discovered via the Anti-de Sitter/Conformal Field Theory correspondence.
We express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the Euler-Lagrange equations of the associated energy functionals. Many solutions are given and discussed. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as …
Study on relative entropy for hypersurfaces in hyperbolic space.
problem Understanding relative entropy for hypersurfaces in hyperbolic space.
method Relate relative entropy to renormalized area and apply monotonicity formula to mean curvature flows.
result Obtained a monotonicity formula for relative entropy in hyperbolic space.
Defines a new geometric quantity for hyperbolic manifolds, showing it's well-defined and invariant.
problem Defining a mass for asymptotically hyperbolic manifolds under weaker conditions.
method Volume-renormalized mass defined as a linear combination of ADM mass and renormalized volume.
result Volume-renormalized mass is well-defined and diffeomorphism invariant under weaker conditions.
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…
Develops methods for computing conformal invariants of submanifolds.
problem Computing conformal invariants of submanifolds.
method Direct construction of extrinsic ambient space, global invariants of conformally compact minimal submanifolds, introduction of conformal submanifold scalars.
result Derives an explicit Gauss--Bonnet--Chern-type formula and proves a rigidity result.
Renormalized volume of minimally bounded regions in asymptotically hyperbolic Einstein spacesmath.DG
Defines renormalized volume for bounded regions in asymptotically hyperbolic Einstein spaces.
problem Calculating the volume of bounded regions in complex geometries.
method Defines renormalized volume, proves Gauss-Bonnet theorem, computes derivative under variations.
result Derives a Gauss-Bonnet theorem for the renormalized volume.
The area renormalization procedure gives an invariant of even-dimensional closed submanifolds in a conformal manifold, which we call the Graham-Witten energy, and it is a generalization of the classical Willmore energy. In this paper, we obtain an explicit formula for the second variation of this energy at minimal subm…
Defines and proves properties of weighted renormalized volume coefficients.
problem None explicitly stated; focuses on mathematical definitions and proofs.
method Defines weighted renormalized volume coefficients and proves their variational nature and polynomial representation.
result Weighted renormalized volume coefficients are variational and can be expressed as polynomials of specific tensors.
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 …
New adapted renormalized volume for hyperbolic 3-manifolds with compressible boundary.
problem Analyzing convex co-compact hyperbolic 3-manifolds with compressible boundaries.
method Defining and analyzing a new version of the renormalized volume.
result The adapted renormalized volume is bounded and has properties analogous to the classical renormalized volume.
The Schwarzian action is linked to the area of Epstein curves in hyperbolic geometry.
problem Understanding the relationship between the Schwarzian action and geometric properties of curves.
method Applying Epstein's construction to relate the Schwarzian action to the area of Epstein curves in hyperbolic disk.
result The Schwarzian action is equivalent to the logarithm of the bi-local observable in Schwarzian field theory.
Proves energy expression on Poincaré-Einstein spaces.
problem Computing renormalized Yang-Mills energy on Poincaré-Einstein manifolds.
method Generalizes Chang-Qing-Yang method for renormalized volumes and uses scattering theory for Schrödinger operators.
result Agrees with anomaly boundary integrand in seven dimensions.
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.
problem Implementing renormalization in neural networks.
method Mapping neural networks to quantum field theory, applying renormalization techniques.
result Changing weight standard deviation corresponds to a renormalization flow.
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 …
New compactification of Teichmüller space via renormalized volume.
problem Compactify Teichmüller space with new distance.
method Horocompactification with renormalized volume.
result Translation length of pseudo-Anosov mapping classes equals hyperbolic volume of mapping tori.
Renormalized pruning improves neural network accuracy.
problem Over-parameterized neural networks waste many parameters.
method Propose renormalizing sparse neural networks.
result Renormalized pruning converges to zero error.
Random harmonic maps into spheres converge to a specific metric under strong convergence of representations.
problem Understanding the behavior of harmonic maps into spheres under representation convergence.
method Introduced renormalized energy and harmonic representatives, proving convergence to a rescaled hyperbolic metric.
result Renormalized energies and harmonic representatives converge to a specific metric under strong convergence of representations.
We compute renormalized curvature integrals on Poincaré-Einstein manifolds.
problem Computing renormalized curvature integrals on Poincaré-Einstein manifolds.
method General procedure that connects Gauss-Bonnet-type formulas and identifies scalar conformal invariants.
result Explicit conformally invariant Gauss--Bonnet-type formulas for compact 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.
problem Calculating volume-renormalized mass for asymptotically hyperbolic manifolds.
method Using Michel's mass invariants and a reduced Hamiltonian perspective, the volume-renormalized mass is deduced.
result The reduced Hamiltonian recovers the volume-renormalized mass and its variations.
The paper studies circle packings using renormalization and subdivision rules.
problem Characterizing and proving properties of circle packings with specific subdivision rules.
method Iterations of skinning maps on Teichmüller spaces, renormalization theory, subdivision rules.
result Uniformly contracting renormalization operator and geometric inflexibility of circle packings.
Volume renormalization for conformally compact asymptotically hyperbolic manifolds in dimension fourmath.DG
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…
Isoperimetric interpretation for the renormalized volume of convex co-compact hyperbolic 3-manifoldsmath.DG
The renormalized volume is reinterpreted using isoperimetric profiles.
problem Understanding the renormalized volume of convex co-compact hyperbolic 3-manifolds.
method Using isoperimetric profiles and Minkowski inequalities.
result A sharp Minkowski inequality for horospherically convex sets in .
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.
problem Defining new CR invariants via renormalized characteristic forms.
method Introducing new curvatures for each renormalized characteristic form.
result The new curvatures' integrals match CR invariants constructed by Marugame.
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…
A method to analyze maps into circles with singularities.
problem Analyzing maps with singularities into circles.
method Renormalization of Dirichlet Lagrangian for -harmonic maps.
result Applications in Willmore energy and frame energies.
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.
problem Computing renormalized volume for knot embeddings in 3-sphere.
method Renormalizing volume associated to singular Yamabe metric.
result Renormalized volume is a global conformal invariant for knots in 3-sphere.
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…