The paper constructs new bimetric conformal invariants using metric perturbations.
problem Developing new conformal invariants in Riemannian geometry.
method Using linear metric perturbations and conformal invariants.
result New bimetric conformal invariants on 4D manifolds are derived.
The paper proves new theorems about specific types of operator perturbations.
problem Analyzing conformal perturbations of Dirac and signature operators.
method Developed Kastler-Kalau-Walze type theorems for specific operator types.
result Established new theorems for six-dimensional manifolds with boundary.
The paper proves two theorems for modified Novikov operators under conformal perturbations.
problem Proving theorems for modified Novikov operators under conformal perturbations.
method Two Kastler-Kalau-Walze type theorems for conformal perturbations of modified Novikov Operators on 4D and 6D compact manifolds.
result Obtained two Kastler-Kalau-Walze type theorems for conformal perturbations of modified Novikov Operators.
Proves K-K-W type theorems for specific types of operators.
problem Analyzes conformal perturbations of twisted Dirac operators.
method Uses Kastler-Kalau-Walze type theorems for four-dimensional manifolds.
result Establishes theorems for both with and without boundary conditions.
In this short note, we prove that conformal classes which are small perturbations of a product conformal class on a product with a standard sphere admit a metric extremal for some Laplace eigenvalue. As part of the arguments we obtain perturbed harmonic maps with constant density.
The paper proves new theorems for Dirac operators on even-dimensional manifolds with boundary.
problem Proving theorems for Dirac operators on manifolds with boundary.
method Establishing general Kastler-Kalau-Walze type theorems for conformal perturbations of Dirac operators.
result Proof of new theorems for Dirac operators on even-dimensional manifolds with boundary.
We prove that in many cases the existence of an extremal metric for some Laplace eigenvalue in a conformal class allows to find extremal metrics in conformal classes close by. As a consequence and as part of the arguments we obtain perturbed harmonic maps with constant density.
The paper studies heat kernels on modified manifolds and bounds their properties.
problem Bounding heat kernels on modified Riemannian manifolds.
method Derives upper bounds and gradient estimates for the heat kernel of (M,ildeg). result Establishes upper bounds and gradient estimates for the heat kernel of modified manifolds.
This paper improves conformal prediction to be robust to perturbations.
problem Ensuring robustness of conformal prediction to natural and adversarial perturbations.
method Probabilistically robust conformal prediction (PRCP) and its adaptive version (aPRCP).
result aPRCP achieves better trade-offs between nominal performance and robustness.
We compute the Ricci curvature of a curved noncommutative three torus. The computation is done both for conformal and non-conformal perturbations of the flat metric. To perturb the flat metric, the standard volume form on the noncommutative three torus is perturbed and the corresponding perturbed Laplacian is analyzed.…
Paper proves existence of solutions for a specific system.
problem Existence of solutions for a conformal Dirac-Einstein system.
method Perturbation methods to prove existence of solutions.
result Existence of solutions for the conformal Dirac-Einstein system.
The conformal Willmore functional (which is conformal invariant in general Riemannian manifold (M,g)) is studied with a perturbative method: the Lyapunov-Schmidt reduction. Existence of critical points is shown in ambient manifolds (R3,gε) -where gε is a metric close and asymptotic to the euclidean o…
Conformal-DP improves differential privacy on manifold data by calibrating perturbations based on local densities.
problem Lack of density-awareness in existing differential privacy mechanisms for manifold data leads to biased and suboptimal privacy-utility trade-offs.
method Proposes Conformal-DP, a density-aware differential privacy mechanism using conformal transformations to calibrate perturbations based on local densities.
result Demonstrates improved privacy-utility trade-off in heterogeneous data distribution settings compared to state-of-the-art mechanisms.
Constructs perturbed Fefferman spaces on almost CR manifolds.
problem Characterize and construct conformal structures on almost CR manifolds.
method Introduces perturbations of Fefferman spaces using semi-basic one-forms.
result Derives conditions for conformally flat spaces on zero sets of almost Einstein scales.
In this paper we prescribe a fourth order conformal invariant on the standard n−sphere, with n≥5, and study the related fourth order elliptic equation. We first find some existence results in the perturbative case. After some blow up analysis we build a homotopy to pass from the perturbative case to the non-pert…
Generalizes Fefferman's structure to CR three-manifolds with additional data.
problem Finding conditions for conformal isometry and existence of metrics.
method Introduces perturbations of Fefferman's conformal circle bundle and investigates existence of metrics.
result Provides conditions for existence of metrics satisfying Einstein equations.
Study robustness of split conformal prediction under adversarial attacks.
problem Ensuring distribution-free coverage guarantees in CP under adversarial conditions.
method Theoretical analysis and extensive experiments on split conformal prediction robustness.
result Prediction coverage varies with calibration-time attack strength, enabling control over coverage under adversarial tests.
Study curvatures on compact pseudo-Hermitian manifolds using special methods.
problem Prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds.
method Upper and lower solutions, perturbation theory of self-adjoint operators, CR conformal deformations.
result Described sets of Webster scalar curvature functions that can be realized.
Deep neural networks improve free energy calculations for peptide conformations.
problem Challenges in developing suitable mappings for free energy perturbation.
method Adapted machine learning approach to train deep neural networks for mapping between Boltzmann distributions.
result Accurate free energy differences calculated between thermodynamic states with spring centers separated by 1 Å and sometimes 2 Å.
We study several problems concerning conformal transformation on metric measure spaces, including the Sobolev space, the differential structure and the curvature-dimension condition under conformal transformations. This is the first result about preservation of lower curvature bounds under perturbation, which is new ev…
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.
The paper is devoted to the study of BRST charge in perturbed two dimensional conformal field theory. The main goal is to write the operator equation expressing the conservation law of BRST charge in perturbed theory in terms of purely algebraic operations on the corresponding operator algebra, which are defined via th…
Kahler geometry explains decoupling of Kerr perturbations.
problem Decoupling of curvature scalars in Kerr spacetime.
method Hidden Kahler structure in Kerr spacetime, showing decoupling as a consequence of Kahler geometry.
result Decoupling of Teukolsky equations on Kahler background.
This paper improves conformal prediction for robust interval estimation under distribution shifts.
problem Robustness of conformal prediction under distribution shifts.
method Modeling distribution shifts using Levy-Prokhorov (LP) ambiguity sets, which capture both local and global perturbations.
result Constructs robust conformal prediction intervals that remain valid under distribution shifts.
Among all conformal classes of Riemannian metrics on CP2, that of the Fubini-Study metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the Seiberg-Witten equations, also yields new results on the total scalar curvature of almost-Kähler 4-manifolds.
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…
Axisymmetric Ricci solitons are rigid under non-axisymmetric perturbations.
problem Understanding the rigidity of axisymmetric Ricci solitons under perturbations.
method Examined non-axisymmetric perturbations of axisymmetric toric Einstein manifolds and Ricci solitons, establishing a rigidity result.
result Axisymmetric Ricci solitons do not admit constant-angle non-axisymmetric perturbations except for conformally flat cases.
The paper solves a curvature problem on a ball's surface near constant values.
problem Prescribing almost constant curvatures on a manifold with boundary.
method Perturbative approach and ansatz by Han and Li.
result New existence results for conformal metrics when curvatures are near constants.
The paper examines stability of Yamabe boundary problem under perturbations.
problem Stability of Yamabe boundary problem under perturbations of mean curvature and scalar curvature.
method Analyzes stability of the Yamabe boundary problem with respect to perturbations of mean curvature and scalar curvature.
result The stability of the Yamabe boundary problem is proven under perturbations from below, but not from above.
Connects robust optimization to conformal prediction for uncertainty sets.
problem Decision-making under uncertainty in sensitive data.
method Defines Mahalanobis distance as a conformity score and generates conformal uncertainty sets.
result Conformal uncertainty sets provide valid and conservative ellipsoidal regions.
We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent σ∈(1/2,1). This extends the result of Chen-Xu (Invent. Math. 187, no. 2, 395-506, 2012) for the scalar curvature flow on the standard unit sphere.
In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles…
It is shown in the paper "Variational Properties of the Gauss-Bonnet Curvatures" of M.L. Labbi, that metrics with constant 2k-Gauss-Bonnet curvature on a closed n-dimensional manifold, 1<2k<n, are critical points for a certain Hilbert type functional with respect to volume preserving conformal variations. This motivate…
We construct new examples of Einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted Hölder spaces.
Solves a conjecture using a new formula on conformally Einstein manifolds.
problem Solving a conjecture in conformal geometry.
method Using an Obata type formula established by previous works.
result Solves Hang-Yang conjecture via an Obata-type argument.
Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.
Study magnetic perturbations in Riemannian and Lorentzian Calderón problems.
problem Determining metrics from boundary measurements under magnetic perturbations.
method Runge approximation for Riemannian case, microlocal analysis for Lorentzian case.
result Metrics can be uniquely determined in both Riemannian and Lorentzian cases under specific perturbations.
Study on 4D Einstein manifolds with Kähler conformal geometry.
problem Exploring 4D Poincaré-Einstein manifolds with Kähler metrics.
method Formulated a Dirichlet boundary value problem and established existence and uniqueness theory.
result Existence and uniqueness of new Poincaré-Einstein metrics.
In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as applications, we derive some compactness results under perturbation conditions when the L^2-…
One method of studying the asymptotic structure of spacetime is to apply Penrose's conformal rescaling technique. In this setting, the Einstein equations for the metric and the conformal factor in the unphysical spacetime degenerate where the conformal factor vanishes, namely at the boundary representing null infinity.…
New methods protect privacy while providing accurate prediction sets.
problem Privacy-preserving conformal prediction for untrusted aggregators.
method Two LDP approaches: k-ary randomized response and binary search response.
result Finite-sample coverage guarantees and robust coverage under randomization.
The fractional Yamabe problem, proposed by González-Qing (2013, Anal. PDE) is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the classical Yamabe problem and the boundary Yamabe problem to the realm of nonlocal co…
The paper finds multiple ways a special curvature can blow up in high dimensions.
problem Finding multiple metrics with constant Q-curvature in high dimensions.
method Constructing small perturbations of standard bubbles.
result Infinitely many smooth metrics with the same constant Q-curvature and arbitrarily large energy.
COLEP improves robustness of conformal prediction via probabilistic circuits.
problem Adversarial perturbations can undermine the coverage guarantees of conformal prediction.
method COLEP uses probabilistic circuits to learn and reason about different semantic concepts, providing certifiable coverage guarantees.
result COLEP achieves higher prediction coverage and accuracy than a single model, especially with non-trivial knowledge models.
Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.
problem Investigating properties of Kähler manifolds with specific curvature conditions.
method Conformal perturbation method.
result Established structure and splitting theorems for Kähler manifolds with non-negative mixed curvature.
Study shows strict inequality for Dirac-type equation on spin manifolds, leading to existence results.
problem Investigating strict inequality for a Dirac-type equation on compact spin manifolds.
method Analyzing a generalized conformally invariant equation involving the Dirac operator with a non-linear convolution term.
result Strict inequality holds, except for round sphere conformal cases, providing existence results for a ground state.
The Clifford torus minimizes Willmore energy closely for small perturbations.
problem Finding the closest shape to the Clifford torus under small perturbations of Willmore energy.
method Analyzing integral 2-varifolds with specific properties and showing quantitative closeness to the Clifford torus.
result The support of the varifold is quantitatively close to the Clifford torus after a conformal transformation.
In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constan…