The J-flow is a parabolic flow on Kahler manifolds. It was defined by Donaldson in the setting of moment maps and by Chen as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. It is shown here that under a certain condition on the initial data, the J-flow converges to a critical metr…
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8 results for “lower-boundedness”
In this paper, we generalize Chen-Tian energy functionals to Kähler-Ricci solitons and prove that the properness of these functionals is equivalent to the existence of Kähler-Ricci solitons. We also discuss the equivalence of the lower boundedness of these functionals and their relation with Tian-Zhu's holomorphic inva…
We investigate the Kähler-Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a Kähler-Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy.
Paper develops proper, lower-bounded losses for weakly supervised classification.
problem Weakly supervised classification with corrupted labels.
method Representation theorem for proper losses, derived condition for lower-boundedness, generalized logit squeezing.
result Proper and lower-bounded losses for weak-label learning.
In this paper, we prove the equivalence of the existence of extremal Kahler metrics and the properness of the modified K energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K energy, the infimum of the Calabi energy and the extremal polynomials. In particular, we give an exa…
Lower bound of modified -energy on a Fano manifold with degeneration for Kähler-Ricci solitonsmath.DG
Study lower bounds on modified K-energy on Fano manifolds with Kähler-Ricci solitons.
problem Lower boundedness of modified K-energy on Fano manifolds.
method Extend Tosatti's method to study Fano manifolds with Kähler-Ricci solitons.
result Establish lower bounds on modified K-energy for Kähler-Ricci solitons.
On Perelman's -entropy and Shannon entropy power for super Ricci flows on metric measure spacesmath.DG
The paper extends entropy formulas to super Ricci flows on metric measure spaces.
problem Entropy formulas for super Ricci flows on metric measure spaces.
method Extending Perelman's -entropy and Shannon entropy power to super Ricci flows.
result Equivalence between volume non-local collapsing property and lower boundedness of -entropy on RCD spaces.
The paper proves conditions for infinite lifetime of Brownian motion and regularity of heat flow.
problem Conditions for infinite lifetime of Brownian motion on Riemannian manifolds with bounded Ricci curvature.
method Derives a Bismut-Elworthy-Li derivative formula and proves the equivalence of lower bounded Ricci curvature to the existence of pathwise couplings.
result Conditions on Ricci curvature ensure infinite lifetime of Brownian motion and regularity of heat flow.