Study on soap bubble clusters, showing manifold properties.
problem Understanding the space of planar soap bubble clusters.
method Analysis of soap bubble clusters as generalized Voronoi partitions.
result The space of planar clusters with positive second variation is an n-dimensional manifold.
We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.
Paper derives second variational formula for statistical manifold mappings.
problem Variational formulas for mappings between statistical manifolds.
method Develops second variational formula for harmonic mappings, defines stability, index, and nullity.
result Shows weakly stability for harmonic mappings into statistical manifolds of non-positive curvature.
We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang-Yau quasi-local energy in general relativity. We prove that the functional is positive definite on large coordinate spheres, and more general on nearly roun…
The stability and the index of complete one-sided minimal surfaces of certain three-dimensional Riemannian manifolds with positive scalar curvature are studied.
J.Eells and L. Lemaire introduced k-harmonic maps, and T. Ichiyama, J. Inoguchi and H.Urakawa showed the first variation formula. In this paper, we give the second variation formula of k-harmonic maps, and show non-existence theorem of proper k-harmonic maps into a Riemannian manifold of non-positive curvature (k >= 2)…
Derives a formula for the second variation of the Laplace eigenvalue functional on manifolds.
problem Calculating the second variation of the Laplace eigenvalue functional on closed manifolds.
method Derives a scale-invariant second variation formula for the Laplace eigenvalue functional.
result Proves that the canonical flat metric on a torus is not a maximal point of the functional in its conformal class.
The paper calculates the second variation of energy functions for families of canonically polarized manifolds.
problem Computing the second variation of energy functions for families of canonically polarized manifolds.
method Analyzing the Dirichlet energy of maps between fibers and using harmonic maps.
result The energy function is plurisubharmonic under certain curvature conditions.
Harmonic gauge simplifies geometric analysis of Riemannian metrics.
problem Analyzing the Hilbert-Einstein functional and its stability.
method Developed a harmonic gauge to eliminate divergence terms and induce elliptic structure.
result Positivity of curvature operator implies spectral stability of the functional.
Proves metrics with positive intermediate Ricci curvature on complex manifolds.
problem Establishing metrics with positive intermediate Ricci curvature on complex manifolds.
method Canonical variation and surgery techniques.
result Existence of metrics with positive intermediate Ricci curvature on various examples.
In this note, we compute the second variational formula for the functional ∫Mv(6)(g)dvg, which was introduced by Graham-Juhl and the first variational formula was obtained by Chang-Fang. We also prove that Einstein manifolds (with dimension ≥7) with positive scalar curvature is a strict local maximum wi…
Proves convexity of minimizers in energy functions with convex potentials.
problem Connectedness and convexity of minimizers in energy functions involving surface tensions and convex potentials.
method Introduces a 'two-point function' to measure lack of convexity and prove negative second variation of the energy.
result Positively answers an old question of Almgren about connectedness and convexity of minimizers.
In Heisenberg group, certain graphs are stable under contact variations but not area-minimizing.
problem Characterizing stable sub-Riemannian graphs in the Heisenberg group.
method Analyzing intrinsic graphs of smooth functions in the first Heisenberg group under contact variations.
result Intrinsic graphs of smooth functions are stable points of sub-Riemannian perimeter under contact variations, but not area-minimizing.
Study stability of compact Ricci solitons using entropy variations.
problem Linear stability condition for compact shrinking Ricci solitons.
method Second variation of Perelman's ν-entropy.
result Necessary and sufficient condition for linear stability.
In this paper we demonstrate that under general conditions there exists a metric in the conformal class of an arbitrary metric on a smooth, closed Riemannian manifold of dimension greater than four such that the Q-curvature of the metric is a constant. Existence of solutions is obtained through the combination of var…
New stability theorem for hypersurfaces in Minkowski spaces.
problem Characterize stable hypersurfaces in Minkowski spaces.
method Introduced stability concept and computed second variation formula.
result Compact stable hypersurfaces in Minkowski spaces are homothetic to the boundary of the convex body.
The study classifies equivariant biharmonic maps and proves stability results for certain maps.
problem Classifying and analyzing equivariant biharmonic maps and their stability.
method Generalized biharmonic equation for equivariant maps, improved second variation formula for biharmonic maps.
result No stable proper biharmonic maps with constant square norm of tension field exist from a compact Riemannian manifold into a space form of positive sectional curvature.
We study different notions of Riemannian curvatures: The p-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the (p,q)-curvatures, which incorporate …
The paper proves geodesic convexity and plurisubharmonicity of energy functions on Teichmüller space.
problem Geodesic convexity and plurisubharmonicity of energy functions on Teichmüller space.
method First and second variations of energy function, strict plurisubharmonicity, and convexity proofs.
result Strict plurisubharmonicity of log(E(z)) on Teichmüller space, and convexity of E(t) along Weil-Petersson geodesics.
New diameter estimate on manifolds with positive Bakry-Émery Ricci tensor.
problem Estimating the diameter of manifolds with specific curvature conditions.
method Using generalized mean curvature comparison on excess function.
result Sharper diameter estimate than previous results.
Develops MENT for interpreting and detecting changes in network trajectories.
problem Distortion of network geometry and invalidation of temporal comparisons in dynamic network analysis.
method Develops Multiscale Euclidean Network Trajectories (MENT) framework based on second-moment geometry.
result Validates and interprets network trajectories through isotropic normalization and orthogonal transformations.
Paper calculates second variation of Graham-Witten energy for spheres.
problem Calculating the second variation of Graham-Witten energy.
method Explicit formula for second variation at minimal submanifolds in Einstein manifolds.
result Totally geodesic spheres in unit sphere are critical points with non-negative second variation.
The paper explores parabolic regularity in geometric variational analysis.
problem Developing calculus rules and computation formulas for second-order generalized differential constructions.
method Introducing and applying the concept of parabolic regularity to geometric aspects of second-order variational analysis.
result Established new calculus rules and computation formulas for second-order generalized differential constructions.
We derive the first and second variation formula for the Green's function pole's value of Paneitz operator on the standard three sphere. In particular it is shown that the first variation vanishes and the second variation is nonpositively definite. Moreover, the second variation vanishes only at the direction of confor…
Study the stability of membranes using Helfrich energy and second variation formula.
problem Stability of membranes under various conditions.
method Developed and applied a second variation formula for the Helfrich energy for a class of surfaces.
result Studied the second variation of the area functional for a specific example.
TrustVI is a fast second-order algorithm for black-box variational inference.
problem Efficiently optimizing variational distributions in complex models.
method Trust-region optimization with minibatch reparameterization.
result TrustVI converges faster and finds better variational distributions than alternatives.
Paper studies second variation for L-minimal submanifolds in pseudo-Sasakian manifolds.
problem Analyzing stability of L-minimal submanifolds in pseudo-Sasakian manifolds.
method Provides a second variation formula and applies it to Lorentzian-Sasakian manifolds.
result Relates L-stability of Legendrians in a Sasakian manifold to their stability in an associated Lorentzian-Sasakian structure.
Derives Lagrangian for minimal surfaces, proving tangential variations vanish.
problem Variational calculus for minimal surfaces.
method Lagrangian formulation, pullback covariant derivative, geometric argument.
result Tangential variations vanish for minimal surfaces.
Introduces intrinsic Riemannian cross-covariance for manifold-valued random objects.
problem Covariance estimation for random objects on Riemannian manifolds.
method Defines covariance and correlation via parallel transport.
result Proposed covariance is independent of coordinate choices.
Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry. The progress in the last two decades has become impressive, being especially re…
Formulas for mean curvature in various spaces.
problem Understanding mean curvature in different geometric settings.
method First and second variation formulas for arbitrary variations in Riemannian and sub-Riemannian manifolds.
result Formulas derived for mean curvature in Heisenberg group via approximation.
We show a very simple and general total second variation formula for Perelman's W-functional at arbitrary points in the space of Riemannian metrics. Moreover we perform a study of the properties of the variations of Kähler structures. We deduce a quite simple and general total second variation formula for P…
Paper proposes a second-order method for faster SVI convergence.
problem Poor convergence rate of first-order SVI algorithms.
method Derives Hessian matrix and implements two numerical schemes for efficient second-order SVI.
result Proposed approach achieves faster convergence compared to first-order SVI.
Author presents the second variational formula for statistical biharmonic maps.
problem Developing a formula for statistical biharmonic maps.
method Introduced the second variational formula for the statistical bi-energy functional.
result The second variational formula can be represented using Hessian curvature in Hessian manifolds.
Inspired by the recent work of Physicists Hertog-Horowitz-Maeda, we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admits nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the…
Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.
problem Understanding variational properties of integral invariants defined from the second fundamental form.
method Derive first variational formulae for integral invariants of degree two, show Euler-Lagrange equation for Chern-Federer energy, and provide examples of submanifolds.
result The Euler-Lagrange equation of the Chern-Federer energy functional reduces to a second order PDE.
We compute the second variation of the Ricci expander entropy and briefly discuss the linear stability of compact negative Einstein manifolds.
Sharp curvature condition implies spherical space form structure.
problem Characterizing manifolds with specific curvature properties.
method Proving diffeomorphism and homeomorphism to spherical space forms using curvature conditions.
result Closed manifolds with $4 rac{1}{2}$-positive curvature operator of the second kind are spherical space forms.
Study bounds the index of minimal submanifolds using energy measures and Yang-Mills-Higgs equations.
problem Bounding the index of codimension 2 minimal submanifolds.
method Second inner variation of energy, convergence of energy measures, and stress-energy tensors.
result Bound the Morse index of the submanifold by the index of critical points.
The paper calculates variations of Einstein-Hilbert action on CR manifolds.
problem Variation of the Einstein-Hilbert action in pseudohermitian geometry.
method Computed first and second variations on CR manifolds, characterized critical points as pseudo-Einstein structures, and analyzed second variation on standard spheres.
result In three dimensions, the second variation of the Einstein-Hilbert action on CR structures differs from the Riemannian case due to embeddability.
Extends multivariate regression for tensor-variate data, identifying brain regions and facial characteristics.
problem Challenges in fitting regression models with multivariate responses and covariates.
method Low-rank tensor formats on regression coefficients and tensor-variate normal distribution for errors.
result Maximum likelihood estimators for tensor-on-tensor regression via block-relaxation algorithms.
New mass inequalities and proofs for causal variational principles.
problem Proving new mass inequalities for causal variational principles.
method Proved a new inequality for minimizers of causal variational principles and applied it to prove the positive mass theorem.
result Introduced a positive quasilocal mass and proved new mass inequalities.
Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.
problem Hopf's conjecture on positive sectional curvature and its failure under relaxed conditions.
method Non-trivial extension of the Four Periodicity Theorem to higher degrees.
result Proves positive Euler characteristic for specific manifolds with positive second intermediate Ricci curvature.
Stein variational Newton method accelerates SVGD for faster inference.
problem Efficient nonparametric variational inference.
method Accelerates SVGD by incorporating second-order information and optimal kernel selection.
result Significant computational gains over original SVGD in multiple test cases.
In this paper we provide a detailed proof of the second variation formula, essentially due to Richard Hamilton, Tom Ilmanen and the first author, for Perelman's ν-entropy. In particular, we correct an error in the stability operator stated in Theorem 6.3 of [2]. Moreover, we obtain a necessary condition for linearly …
We consider the second variational derivative of a given gauge-natural invariant Lagrangian taken with respect to (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms. By requiring such a second variational derivative to vanish, {\em via} the Second Noether Theorem we find t…
We show a quite simple second variation formula for Perelman's W-functional along the modified Kähler-Ricci flow over Fano manifolds.
ODE2VAE learns latent dynamics for sequential data.
problem Learning latent dynamics for high-dimensional sequential data.
method Deep generative second order ODE model with Bayesian neural networks.
result State-of-the-art performance in long-term motion prediction and imputation.