We study the negative gradient flow of the spinorial energy functional (introduced by Ammann, Weiß, and Witt) on 3-dimensional Berger spheres. For a certain class of spinors we show that the Berger spheres collapse to a 2-dimensional sphere. Moreover, for special cases, we prove that the volume-normalized standard 3-sp…
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26 results for “volume-normalization”
The Fubini-Study metric minimizes a volume-normalized holomorphic systole in .
problem Finding metrics with minimal holomorphic systoles in complex projective spaces.
method Introduced holomorphic -systole and used Gauduchon metrics to establish minimization.
result The Fubini-Study metric locally minimizes the volume-normalized holomorphic -systole.
Study on normalized Betti numbers in non-positively curved manifolds.
problem Convergence of normalized Betti numbers in non-positively curved manifolds.
method Benjamini-Schramm convergence, volume-normalization, irreducible symmetric spaces of noncompact type.
result Normalized Betti numbers converge for non-compact manifolds.
The paper analyzes systoles of complex projective spaces under various metrics.
problem Behavior of systoles in complex projective spaces for different metrics.
method Integral geometric techniques and careful analysis of systole functional.
result Balanced metrics locally minimize the systole on volume-normalized metrics.
In earlier work, carrying out numerical simulations of the Ricci flow of families of rotationally symmetric geometries on , we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci flow for a ``critical'' initial geometry - one which is at the transition point bet…
We investigate the gradient flow of the norm of the Riemannian curvature on surfaces. We show long time existence with arbitrary initial data, and exponential convergence of the volume normalized flow to a constant scalar curvature metric when the initial energy is below a constant determined by the Euler charact…
Upper bounds for the second nonzero eigenvalue of the Laplacian via folding and conformal volumemath.SP
Upper bound for Laplacian eigenvalue via conformal volume.
problem Finding upper bounds for Laplacian eigenvalues.
method Using conformal volume to derive an upper bound.
result Effective upper bound for Laplacian eigenvalues on manifolds.
New proof shows affine manifolds with parallel volume are Riemannian-flat.
problem Characterize compact affine manifolds with parallel volume.
method Construct a representative metric with Levi-Civita connection, using Hessian of volume-normalized distance functions.
result Affine manifolds with parallel volume are Riemannian-flat.
New Crofton formulae derived from existing ones.
problem Generalizing Crofton formulae for products.
method Calculations in the ring of normal densities.
result Generalizations of Crofton formulae in terms of mixed Riemannian volume.
Study eigenvalues of conformal Laplacian under Sire-Xu normalization.
problem Existence and properties of extremal eigenvalues under a specific normalization.
method Variational analysis of eigenvalue functional under Sire-Xu normalization.
result Necessary conditions and existence results for extremal eigenvalues.
We consider the volume-normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton is a local maximum of Perelman's shrinker entropy, any normalized Ricci flow starting close to it exists for all time and converges towards a Ricci soliton. If is not a local maxim…
We study the behaviour of the Ricci Yang-Mills flow for U(1) bundles on surfaces. We show that existence for the flow reduces to a bound on the isoperimetric constant. In the presence of such a bound, we show that on , if the bundle is nontrivial, the flow exists for all time. For higher genus surfaces the flow al…
Paper solves capillary Orlicz-Minkowski problem with new inequalities.
problem Finding capillary convex bodies with prescribed Orlicz surface area measures.
method Continuity method and inequalities to solve the capillary even Orlicz-Minkowski problem.
result Volume-normalized smooth solutions and inequalities established.
Let be a principal U(1)-bundle over a closed manifold . On , one can define a modified version of the Ricci flow called the Ricci Yang-Mills flow, due to these equations being a coupling of Ricci flow and the Yang-Mills heat flow. We use maximal regularity theory and ideas of Simonett concerning the asymptoti…
We consider some elementary aspects of the geometry of the space of probability measures endowed with Wasserstein distance. In such a setting, we discuss the various terms entering Perelman's shrinker entropy, and characterize two new monotonic functionals for the volume-normalized Ricci flow. One is obtained by a resc…
Introduces a new -Hilbert functional in -geometry.
problem None explicitly stated; focuses on introducing a new functional.
method Inspired by the Einstein-Hilbert functional, defines a new -Hilbert functional on -structures.
result Torsion-free and nearly -structures are saddle critical points of the volume-normalized -Hilbert functional.
We characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for example, this holds when the critical point is non-degenerate), then we show that the…
The paper extends Yamabe flow results to non-compact manifolds with bounded geometry.
problem Yamabe flow convergence issues on manifolds with infinite volume.
method Curvature-normalized Yamabe flow for manifolds with bounded geometry.
result Long-time existence and convergence of the flow for negative scalar curvature.
Reference metrics are used to define the differential structure on multicube representations of manifolds, i.e., they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous. This paper introduces a general procedure for constructing reference metrics…
A new geometric flow -flow on 3-manifolds shrinks or preserves homogeneous spheres.
problem Analyzing the behavior of Thurston's model geometries under the -flow.
method Defining and studying the -flow on 3-dimensional Riemannian manifolds, using a DeTurck-type argument for short-time existence.
result The -flow shrinks or preserves homogeneous spheres, showing short-time existence.
Researchers prove a stability result for a 3-sphere inequality, extending previous work.
problem Quantitative stability of nonlinear Yamabe-type inequalities on the 3-sphere.
method Proved a two-term refinement of the Schur lemma inequality in the conformal class of the 3-sphere.
result Deduced quantitative stability of an entire family of nonlinear Yamabe-type inequalities.
Framework analyzes stock price co-movement with fundamentals using big data.
problem Understanding complex relationships between stock price co-movements and fundamental characteristics.
method Advanced big data techniques, four regression models.
result Identifies leading co-movement stocks and their influencing factors.
Study of limits of Einstein-Bogomol'nyi metrics on P^1 in two regimes.
problem Understanding limits of Einstein-Bogomol'nyi metrics on P^1.
method Analysis of two regimes: dissolving limit and large volume limit.
result Recovery of Einstein-Bogomol'nyi metrics on C with total string number N' for each N'.
Computes canonical heights for arithmetic log surfaces using Hurwitz zeta function.
problem Computing canonical heights for arithmetic log surfaces.
method Introduces a canonical height and uses limits of periods to compute it.
result Explicit formulas for canonical heights of arithmetic log surfaces, including (P_1,D).
Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kähler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsev…
A new coordinate system for SPD matrices simplifies computations and generative modeling.
problem Computing and modeling SPD matrices
method Reverse telescoping coordinate system
result Significantly reduces computational complexity and facilitates generative modeling.