• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1193-1206
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• 2009

C. Guenther studied the stability of DeTurck flows by using maximal regularity theory and center manifolds, but these arguments can not solve the stability of Ricci flows because the Ricci flow equation is not strictly parabolic. Recognizing this deficiency, the present paper considers and obtains the stability of Ricci flows, and of quasi-Ricci flows in view of some Killing conditions.

• East Asian mathematical journal
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• v.37 no.5
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• pp.577-584
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• 2021

Let p : X → D be a proper surjective holomorphic submersion where X is a Kähler manifold and D is the unit disc in ℂ. Let Ω be a d-closed semi-positive real (1, 1)-form on X. If each X_s := p^-1(s) for s ∈ D satisfies $-c_1(X_s)+{\Omega}{\mid}_{X_s}$ is Kähler, then the Kähler-Ricci flow twisted by ${\Omega}{\mid}_{X_s}$ has a long time solution by Cao's theorem. This family of twisted Kähler-Ricci flows induces a relative Kähler form ω(t) on the total space X. In this paper, we prove that the positivity of ω(t) is preserved along the twisted Kähler-Ricci flow.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1361-1371
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• 2017

In this paper, we introduce a framework of (${\alpha},{\beta}$)-flows on triangulated manifolds with two and three dimensions, which unifies several discrete curvature flows previously defined in the literature.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.855-861
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• 2019

The geodesics on the round 2-sphere $S^2$ are all simple closed curves of equal length. In 1903 Otto Zoll introduced other Riemannian surfaces with the same property. After that, his name is attached to the Riemannian manifolds whose geodesics are all simple closed curves of the same length. The question that "whether or not the set of Zoll metrics on 2-sphere $S^2$ is connected?" is still an outstanding open problem in the theory of Zoll manifolds. In the present paper, continuing the work of D. Jane for the case of the Ricci flow, we show that a naive application of some famous geometric flows does not work to answer this problem. In fact, we identify an attribute of Zoll manifolds and prove that along the geometric flows this quantity no longer reflects a Zoll metric. At the end, we will establish an alternative proof of this fact.