We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
Paper proves discrete uniformizations converge to continuous for surfaces of genus ≥1.
problem Computing uniformizations for surfaces of genus >1.
method Discrete conformality and uniformization on triangle meshes.
result Discrete uniformizations approximate continuous uniformization for closed surfaces of genus ≥1.
The paper proves uniformization for specific curvature types on manifolds.
problem Uniformizing fourth order conformal curvature on Riemannian manifolds.
method Proving existence of conformal deformations for specific curvature conditions.
result Existence of conformal deformations for positive Yamabe invariant and total Q-curvature.
Uniform K-stability ensures existence of special metrics on toric manifolds.
problem Existence of conformally Kähler, Einstein-Maxwell metrics on toric manifolds.
method Introducing uniform K-stability and showing its equivalence to properness of relative K-energy.
result Uniform K-stability is necessary and sufficient for the existence of f-extremal metrics on toric manifolds. Conformal Test Martingales can be 'blind' to significant changes in data distribution.
problem The converse of exchangeability does not hold, leading to potential blindness of CTMs.
method Explicit construction of A-cryptic change-point using bivariate Gaussian distributions. result CTMs can be perfectly cryptic to a significant change in marginal means.
New proof of uniformization for hyperbolic foliations.
problem Uniformization of foliated spaces by surfaces of hyperbolic type.
method Laminated Ricci flow to find a conformally equivalent metric with constant curvature -1.
result Existence of a laminated Riemannian metric with leaves of constant Gaussian curvature -1.
In this paper a new connection between the discrete conformal geometry problem of disk pattern construction and the continuous conformal geometry problem of metric uniformization is presented. In a nutshell, we discuss how to construct disk patterns by optimizing an objective function, which turns out to be intimately …
The paper derives uniform stability-based coverage bounds for conformal prediction methods.
problem Establishing theoretical guarantees for conformal prediction methods.
method Uniform stability perspective applied to full-conformal, jackknife+, and CV+ prediction regions.
result Coverage bounds for finite-dimensional models derived using a concentration argument.
Solves modified Schouten tensor problems in conformal metric classes.
problem Prescribed problems for modified Schouten tensors in conformal classes of metrics.
method Uniform ellipticity confirmation under topological and functional constraints.
result Extends results from previous work on smooth complete metrics.
Length metrics can be closely approximated by conformally flat metrics.
problem Approximating length metrics with conformally flat metrics.
method Uniform approximation of length metrics by conformally flat Riemannian metrics.
result Any length metric on \(\mathbb{R}^d\) can be uniformly approximated by conformally flat Riemannian metrics.
Validates conformal prediction for network data under non-uniform sampling.
problem Validity of conformal prediction for network data under non-representative sampling.
method Interprets sampling mechanisms as selection rules, studies validity conditional on selection events, uses permutation invariance and joint exchangeability.
result Finite-sample validity of conformal prediction for certain selection events and asymptotic validity for random walk sampling.
Paper shows no finite time singularities for smooth conformal heat flow of harmonic maps.
problem Smoothness of conformal heat flow of harmonic maps.
method Combines harmonic map flow with metric evolution in conformal direction.
result No finite time singularity occurs for the flow, and under certain conditions, maps converge to a point.
Uniform estimates for elliptic problems near polygonal domains.
problem Proving uniform solvability estimates for elliptic problems near polygonal domains.
method Suitable conformal modification of the metric to make the union of domains a manifold with boundary and relative bounded geometry.
result Rounding off the corners of the limit polygonal domain.
We prove effective uniformization for nearly round 2-spheres and investigate their stability.
problem Proving effective uniformization for nearly round 2-spheres and their stability.
method Utilizing an identity related to the third-order differential of the conformal factor, and an isometric embedding of a round sphere into Euclidean space using an orthogonal basis of the first eigenspace of the Laplacian operator.
result We provide a simplified proof of effective uniformization and its stability.
Uniformizes surfaces with boundaries, focusing on triple junctions.
problem Uniformization of surfaces with boundaries, especially triple junctions.
method Extends conformal structure results to triple junction surfaces.
result Weak uniformization results for triple junction surfaces.
Discrete conformal maps on surfaces with vertex decorations are studied.
problem Discrete conformal equivalence for decorated piecewise Euclidean surfaces.
method Intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces, and convex hyperbolic polyhedra; concave variational principle.
result Proof of discrete uniformization theorem for decorated PE-surfaces.
Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler.
problem Chern version of the constant holomorphic sectional curvature conjecture for compact locally conformal Kähler manifolds
method Prove the conjecture using curvature identities and properties of Kähler metrics
result Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler
The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.
Uniform convergence of metrics on surfaces with bounded curvature measures proved.
problem Proving uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature.
method Weak convergence of measures and analytic approximation of metrics.
result Uniform convergence of metrics on Alexandrov surfaces proved.
We give some uniform estimates for constant mean curvature solutions of the conformal vacuum Einstein constraint equations on compact manifolds. Existence of those solutions was given in a paper by J. Isenberg.
Noise-Aware Conformal Prediction (NACP) calibrates CP for noisy labels.
problem Calibrating Conformal Prediction with noisy labels.
method Estimate conformal threshold from noisy labels using uniform noise coverage guarantee.
result Finite sample coverage guarantee for uniform noise remains effective in high-class tasks.
The purpose of this paper is to introduce a geometric structure called pseudo-conformal quaternionic CR structure on a (4n+3)-dimensional mamnifold and then exhibit a quaternionic analogue of Chern-Moser's CR structure and uniformization.
Paper generalizes discrete uniformization for genus-zero surfaces.
problem Discrete uniformization for surfaces of genus zero.
method Reduction to planar cases via stereographic projections.
result Generalization of discrete uniformization to genus-zero surfaces.
Harmonic maps from surfaces to CAT(k) spheres are branched coverings.
problem Uniformization of surfaces with CAT(k) metrics.
method Almost conformal harmonic maps and branched coverings.
result CAT(k) spheres are conformally equivalent to the 2-sphere.
We study the equation Δgu−4(n−1)n−2R(g)u+Kup=0(1+ζ≤p≤n−2n+2) on locally conformally flat compact manifolds (Mn,g). We prove the following: (i) When the scalar curvature R(g)>0 and the dimension n≥4, under suitable conditions on K, all positive solutions u have uniform u…
Survey on uniformization of metric surfaces, including fractal and topological manifolds.
problem Uniformization of metric surfaces homeomorphic to 2D topological manifolds.
method Various uniformization theorems, including quasisymmetric and quasiconformal approaches.
result Uniformization results for metric spheres and arbitrary metric surfaces.
Defines W-volume for planar domains with circular boundaries, relating to Laplacian determinant and Schottky uniformization.
problem Determining the volume of conformal metrics on planar domains with circular boundaries.
method Extending Epstein maps to conformal metrics, defining W-volume, using Schottky uniformization and Loewner energy.
result Shows a bound on the renormalized volume of Schottky uniformization and provides a realization of Loewner energy.
In this thesis a connection between the worlds of discrete and continuous conformal geometry is explored. Specifically, a disk pattern production theroem is proved using an energy which measures how ``uniform'' the angle data of a triangulation is, see also math.DG/0002150. Then this energy is averaged over all the Del…
We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization the…
The paper proves a theorem for discretizing Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with nonpositive Euler number.
method Discrete conformal theory and variational principles with constraints.
result Each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a metric with a specific discrete curvature constant.
A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying …
In this paper, we prove a uniform approximation theorem with interpolation for complete conformal minimal surfaces with finite total curvature in the Euclidean space Rn (n≥3). As application, we obtain a Mittag-Leffler type theorem for complete conformal minimal immersions M→Rn on any ope…
Study infinite genus surfaces and Schottky groups for uniformization.
problem Investigate infinite genus surfaces and Schottky groups for uniformization.
method Definitions and proofs for infinite genus surfaces and Schottky groups, showing uniformization by Schottky groups.
result Infinite genus surfaces and handlebodies can be topologically and quasiconformally uniformized by Schottky groups.
A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature me…
Quasispheres can be approximated by smooth spheres.
problem Characterizing quasispheres using geometric conditions.
method Proving every quasisphere is a limit of smooth spheres and providing necessary and sufficient conditions for uniform quasispheres.
result Every quasisphere can be approximated by uniform quasispheres that satisfy specific geometric conditions.
Motivated by a recent work of X. Chen and M. Zhu (Commun. Math. Stat., 1 (2013) 369-385), we establish a Trudinger-Moser inequality on compact Riemannian surface without boundary. The proof is based on blow-up analysis together with Carleson-Chang's result (Bull. Sci. Math. 110 (1986) 113-127). This inequality is diffe…
The paper develops p-values for outlier detection using conformal inference.
problem Detecting outliers in nonparametric data sets.
method Conformal inference framework for constructing marginally valid but mutually dependent p-values.
result Valid p-values for outlier detection with conditional independence and marginal false discovery rate control.
We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we obtain some geometric applications. Among them, we emphasize the following ones: 1. A Gunning-Narasimhan type theorem for non-orientable conformal surfaces. 2. An …
Method computes harmonic and conformal maps from point clouds.
problem Computing maps from irregular point cloud data.
method Meshless method using cubic lattice approximations.
result Harmonic and conformal maps computed accurately.
Researchers solve metric curvature equations on manifolds with boundary.
problem Finding complete conformal metrics with specific curvature functions.
method Revealed algebraic structure of fully nonlinear equations; used topological obstructions.
result Solved a class of fully nonlinear equations for conformal metrics.
Study shows limits of metrics with positive scalar curvature on spheres.
problem Non-negativity of scalar curvature is not preserved under certain limits.
method Examined metrics conformal to the round metric on Sn for n≥4. result Any conformal metric to the round metric on Sn for n≥4 can be a limit of metrics with positive scalar curvature. Proves transitivity of real Anosov diffeomorphisms with specific properties.
problem Transitivity of real Anosov diffeomorphisms with specific properties.
method Proves transitivity using specific properties of real Anosov diffeomorphisms.
result Proves transitivity of real Anosov diffeomorphisms.
We consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the Ricci tensor to be a constant. It is related to the notion of minimal volumes in comparison geometry. Such a metric with the smallest eigenvalue of the Ricci tensor to be a constant is an extremal metric of volume in…
We show here that the Nielsen core of the bumping set of the domain of discontinuity of a Kleinian group Γ is the boundary of the characteristic submanifold of the associated 3-manifold with boundary. Some examples of interesting characteristic submanifolds are given. We also give a construction of the characteristic…
We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu−e2u=K0(z) on general open surfaces. A few oth…
The Gursky-Streets equation are introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of σ2 Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equations with uniform C1,1 estima…
Study fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.
problem Finding conformal Hermitian metrics with prescribed curvature functions.
method Blow-up argument and partial uniform ellipticity.
result Our assumptions are almost sharp, with some geometric function theory obstructions.
Develops anytime-valid conformal and PAC prediction for streaming data.
problem Lack of guarantee in traditional conformal methods for sequential settings.
method Extends conformal and PAC prediction frameworks to handle streaming data.
result Provides anytime-valid prediction sets for sequential settings.