Proposes a method to improve class-conditional conformal prediction for many classes.
problem Weak guarantees for specific classes in classification problems.
method Clusters classes with similar conformal scores and performs conformal prediction at the cluster level.
result Clustered conformal typically outperforms existing methods in class-conditional coverage and set size metrics.
Extremal metrics found for Laplace eigenvalues in nearby conformal classes.
problem Finding extremal metrics for Laplace eigenvalues in perturbed conformal classes.
method Existence of extremal metrics in a conformal class allows finding similar metrics in nearby classes.
result Perturbed harmonic maps with constant density obtained as part of the arguments.
Simple conformally recurrent spaces are identified as pp-waves.
problem Characterizing conformally recurrent space-times.
method Analyzing dimension n>3 space-times.
result Simple conformally recurrent space-times are conformally recurrent pp-waves.
The study preserves upper bounds of total scalar curvature in conformal classes.
problem Preserving upper bounds of total scalar curvature in conformal classes.
method Analyzing Yamabe constant and scalar curvature conditions.
result The upper bound condition of total scalar curvature is preserved in a conformal class.
Study of Lorentzian surfaces with Killing fields, characterizing their conformal classes.
problem Characterizing conformal classes of Lorentzian surfaces with Killing fields.
method Defining a map associating conformal classes to vector fields on the circle, analyzing finite-dimensional fibers.
result Finite-dimensional fibers of the map, allowing characterization of conformal classes.
The study shows that certain conformal classes on S^7 cannot be conformal infinities of Poincaré-Einstein metrics.
problem Existence of conformal classes on S^7 that cannot be conformal infinities of Poincaré-Einstein metrics.
method Proved the existence of infinitely many positive conformal classes on S^7 that cannot be conformal infinities of Poincaré-Einstein metrics on the ball B^8.
result Proved a sharp inequality between the Yamabe invariant of the conformal infinity and the Yamabe invariant of the interior.
New metrics found for product spaces with slight changes.
problem Finding metrics for Laplace eigenvalues in perturbed conformal classes.
method Proved existence of extremal metrics for Laplace eigenvalues in perturbed product conformal classes.
result Existence of metrics extremal for some Laplace eigenvalues in perturbed product conformal classes.
We prove that the conformal immersions of complex two tori into S3 which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fi…
New order defined for conformal classes, impacts Bartnik's conjecture.
problem Bartnik's conjecture and its implications under different energy conditions.
method Defined a new order on conformal classes and analyzed implications under null energy condition.
result The null energy condition can lead to future complete metrics in any dimension.
New conformal prediction methods for long-tailed classification problems.
problem Rare classes are systematically omitted in existing conformal prediction methods.
method Introduced a new conformal score function and a new interpolation procedure.
result Smoothly trade off set size and class-conditional coverage.
The largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric, is the class of the conformal Riemannian P-manifolds. This class is an analogue of the class of the conformal Kähler manifolds in almost Hermitian geometry. The …
Proves Lorentzian manifold properties for analytic 3D spaces.
problem Analyzing properties of Lorentzian manifolds.
method Analyzes compact, real-analytic, three-dimensional Lorentzian manifolds.
result Identity conformal group preserves metric or manifold is flat.
Researchers solve a complex problem about determining Riemannian manifolds.
problem Determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map.
method Uses conformal invariance and introduces a new coordinate system to solve the problem on a real-analytic Riemannian manifold.
result Locally conformally real-analytic manifolds in dimensions ≥ 3 can be determined from the Dirichlet-to-Neumann map.
Generalizes classical construction to conformal structures.
problem Characterize and describe conformal structures from projective classes.
method Generalizes Patterson-Walker construction to split-signature conformal structures.
result Complete description of Einstein metrics and all symmetries.
Study projective and almost conformally symplectic structures on manifolds.
problem Relations between projective and almost conformally symplectic structures.
method Single almost conformally symplectic connection with totally trace-free torsion.
result Generalizes Fedosov structures and encodes variability of connections in projective class.
Study of Steklov eigenvalues on degenerating conformal classes.
problem Understanding Steklov eigenvalues on surfaces with boundaries.
method Precise formula for the limit of Steklov eigenvalues on degenerating conformal classes.
result The limit of Steklov eigenvalues equals 2πk for surfaces with boundaries. Compact metrics with similar scalar curvature are found in a specific 4-manifold class.
problem Finding metrics with similar scalar curvature in a conformal class.
method Proving compactness of metrics isospectral to a given metric with specific curvature properties.
result The set of isospectral metrics is compact in the C∞ topology. New metric proves uniqueness of solutions to a specific 4-manifold problem.
problem Uniqueness of solutions to the σ2-Yamabe problem on 4-manifolds. method Defined a new formal Riemannian metric on conformal classes.
result Solutions are unique unless the manifold is round.
Classifies conformal transformations in spacetimes without observer horizons.
problem Understanding conformal transformations in spacetimes without observer horizons.
method Proves classification of conformal transformations into two types: escaping and non-escaping.
result Conformal transformations of Einstein's static universe are classified.
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.
Investigates max width of unit volume 3-spheres in conformal classes.
problem Maximizing width of 3-spheres with fixed volume and conformal class.
method Simon-Smith min-max theory applied to Riemannian 3-spheres.
result Characterizes maximising metrics and how they change with conformal class.
Study on 3-manifolds finds regular conformal metrics for rough metrics.
problem Characterize conformal metrics for rough Riemannian metrics on 3-manifolds.
method Analogous to the Yamabe problem, study conformal classes and regularity.
result Characterize when a more regular representative exists in the conformal class.
Survey about extremum of eigenvalues of geometric operators within a conformal class of a compact riemannian manifold.
We consider the class of all conformal mappings from a compact Riemann surface into the threedimensional or fourdimensional Euclidean space. A sequence in this class with bounded Willmore functional is shown to have a sequence of conformal transformations of the target space, such that a subsequence of the transformed …
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.
Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.
problem Finding conditions for conformal deformations to constant scalar curvature in conic metrics.
method Analyzes conformal deformations within a class of incomplete Riemannian metrics that generalize conic orbifold singularities.
result Determines sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1.
Paper studies conformal vector fields in specific Finsler spaces.
problem Characterizing and understanding conformal vector fields in (α,β) spaces. method Using principles to characterize and determine local structure of conformal vector fields under curvature conditions.
result Construction of non-homothetic conformal vector fields on locally projectively Randers spaces.
By a conformal string in Euclidean space is meant a closed critical curve with non-constant conformal curvatures of the conformal arclength functional. We prove that (1) the set of conformal classes of conformal strings is in 1-1 correspondence with the rational points of the complex domain $\{q\in \mathbb{C} \,:\, 1/2…
Paper connects hypersurfaces in 4D to surfaces in 3-sphere.
problem Understanding conformally flat hypersurfaces in 4D space forms.
method Characterizes conformal structures and relates to surfaces in 3-sphere.
result Relates 2-metrics in 3-sphere to surfaces giving rise to conformally flat hypersurfaces.
In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and a diameter bound f…
The study proves properties of metrics and their conformal classes on specific manifolds.
problem Understanding metrics and their conformal classes on certain manifolds.
method Combining Simons' gap theorem with minimal isometric embeddings and coherent embeddings of standard Einstein metrics.
result The conformal classes of the product metrics and projective spaces realize the sigma invariant uniquely.
Study on flexibility of metric entropy and geodesic flow constraints on surfaces.
problem Flexibility and constraints of metrics and flows on surfaces.
method Analysis of geodesic flow, entropy, and metrics in a fixed conformal class.
result Additional restrictions on topological entropy and new geometric properties.
For G_2-manifolds the Fernández-Gray class X_1+X_4 is shown to consist of the union of the class X_4 of G_2-manifolds locally conformal to parallel G_2-structures and that of conformal transformations of nearly parallel or weak holonomy G_2-manifolds of type X_1. The analogous conclusion is obtained for Gray-Hervella c…
Improves conformal prediction by combining multiple score functions and optimizing weights.
problem Limitations of single-score conformal predictors in multi-class classification.
method Combines multiple score functions and optimizes weights to minimize prediction set size.
result Consistently outperforms single-score conformal predictors while maintaining valid coverage.
Paper proves uniqueness of Einstein metrics on balls.
problem Uniqueness of conformally compact Einstein metrics on balls.
method Compactness result for conformally compact Einstein metrics.
result Global uniqueness of Graham-Lee metrics on balls.
We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal …
The conformal module of conjugacy classes of braids implicitly appeared in a paper of Lin and Gorin in connection with their interest in the 13. Hilbert Problem. This invariant is the supremum of conformal modules (in the sense of Ahlfors) of certain annuli related to the conjugacy class. This note states that the conf…
The paper develops distribution-free methods for ordinal classification.
problem Constructing valid prediction sets for ordinal classification problems.
method Leveraging conformal prediction and multiple testing with FWER control.
result The proposed methods achieve satisfactory levels of marginal and class-specific conditional coverages.
The class W_1 of conformal Riemannian P-manifolds is the largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric. This class is an analogue of the class of conformal Kaehler manifolds in almost Hermitian geometry. In the p…
Consider an asymptotically flat Riemannian manifold (M,g) of dimension n≥3 with nonempty compact boundary. We recall the harmonic conformal class [g]h of the metric, which consists of all conformal rescalings given by a harmonic function raised to an appropriate power. The geometric significance is that eve…
The study proves a new upper bound for isoperimetric ratio in scalar-flat conformal classes.
problem Finding the supremum of isoperimetric ratio over scalar-flat conformal classes.
method Analyzing the supremum of isoperimetric ratio over scalar-flat conformal classes with specific conditions.
result The supremum of the isoperimetric ratio is strictly larger than the Euclidean best constant and is achieved under certain conditions.
Boundary distances determine conformal metrics
problem Determining conformal metrics from boundary distances
method Comparing renormalized boundary distances
result Metrics are equal if distances match
Conformal prediction fails to cover minority classes in imbalanced datasets, but a class-conditional fix improves coverage.
problem Conformal prediction fails to cover minority classes in imbalanced datasets, leading to poor performance on rare labels.
method Class-conditional conformal prediction to improve coverage of minority classes.
result Class-conditional conformal prediction restores minority coverage to target with a modest increase in prediction-set size.
New framework for conformal equivariant cycles in KK-theory.
problem Tackles conformal equivariance in unbounded KK-theory.
method Extends unbounded Kasparov theory with novel perturbation theory.
result Defines new unbounded representatives of Kasparov classes.
Canonical metrics and conformal invariants are presented for closed oriented even-dimensional manifolds with non-degenerate conformal structures and in particular for compact Riemann surfaces.
For a generic distribution of rank two on a manifold M of dimension five, we introduce the notion of a generalized contact form. To such a form we associate a generalized Reeb field and a partial connection. From these data, we explicitly constructed a pseudo--Riemannian metric on M of split signature. We prove tha…
This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some coni…
The paper finds new constrained Willmore minimizers for non-rectangular tori.
problem Finding constrained Willmore minimizers for non-rectangular tori.
method Analyzing immersed tori in 3-space to minimize Willmore energy.
result The candidates constructed in previous work are constrained Willmore minimizers in certain non-rectangular conformal classes.