In this paper, we study gradient solitons to the Ricci flow coupled with harmonic map heat flow. We derive new identities on solitons similar to those on gradient solitons of the Ricci flow. When the soliton is compac...In this paper, we study gradient solitons to the Ricci flow coupled with harmonic map heat flow. We derive new identities on solitons similar to those on gradient solitons of the Ricci flow. When the soliton is compact, we get a classification result. We also discuss the relation with quasi-Einstein manifolds.展开更多
In this paper,we construct an ancient solution by using a given shrinking breather and prove a no shrinking breather theorem for noncompact harmonic Ricci flow under the condition that Sic:=Ric−α∇φ⊗∇φis bounded fro...In this paper,we construct an ancient solution by using a given shrinking breather and prove a no shrinking breather theorem for noncompact harmonic Ricci flow under the condition that Sic:=Ric−α∇φ⊗∇φis bounded from below.展开更多
Let Mn be an embedded closed submanifold ofR^(k+1) or a smooth bounded domain inR_(n),where n≥3.We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time,provi...Let Mn be an embedded closed submanifold ofR^(k+1) or a smooth bounded domain inR_(n),where n≥3.We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time,provided that the initial map u0 is in a suitable nontrivial homotopy class with energy small enough.展开更多
The success of hydrodynamics in high energy heavy-ion collisions leads to a flow paradigm, to understand the observed features of harmonic flow in terms of the medium collective expansion with respect to initial state...The success of hydrodynamics in high energy heavy-ion collisions leads to a flow paradigm, to understand the observed features of harmonic flow in terms of the medium collective expansion with respect to initial state geometrical properties. In this review, we present some essential ingredients in the flow paradigm, including the hydrodynamic modeling, the characterization of initial state geometry and the medium response relations. The extension of the flow paradigm to small colliding systems is also discussed.展开更多
The partial regularity of the weak heat flow of harmonic maps from a Riemannian manifold Al with boundary into general compact Riemannian manifold N without boundary is consid-ered. It is shown that the singular set S...The partial regularity of the weak heat flow of harmonic maps from a Riemannian manifold Al with boundary into general compact Riemannian manifold N without boundary is consid-ered. It is shown that the singular set Sing(u) of the weak heat flow satisfies H(Sing(u)) 0, with is = dimensionM. Here is Hausdorff measure with respect to parabolic metric ρ(x,t),(y,s)=max{|x-y|, }.展开更多
基金supported by NSFC(Grant No.11171143)Zhejiang Provincial Natural Science Foundation of China(Project No.LY13A010009 and LY14A010021)supported by the Fonds National de la Recherche Luxembourg(OPEN Project GEOMREV)
文摘In this paper, we study gradient solitons to the Ricci flow coupled with harmonic map heat flow. We derive new identities on solitons similar to those on gradient solitons of the Ricci flow. When the soliton is compact, we get a classification result. We also discuss the relation with quasi-Einstein manifolds.
文摘In this paper,we construct an ancient solution by using a given shrinking breather and prove a no shrinking breather theorem for noncompact harmonic Ricci flow under the condition that Sic:=Ric−α∇φ⊗∇φis bounded from below.
基金supported partially by NSFC(Grant Nos.12141103 and 12301074)Guangzhou Basic and Applied Basic Research Foundation(Grant No.2024A04J3637)+1 种基金supported partially by NSFC(Grant No.11971400)National Key Research and Development Projects of China(Grant No.2020YFA0712500)。
文摘Let Mn be an embedded closed submanifold ofR^(k+1) or a smooth bounded domain inR_(n),where n≥3.We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time,provided that the initial map u0 is in a suitable nontrivial homotopy class with energy small enough.
基金Supported by Natural Sciences and Engineering Research Council of Canada
文摘The success of hydrodynamics in high energy heavy-ion collisions leads to a flow paradigm, to understand the observed features of harmonic flow in terms of the medium collective expansion with respect to initial state geometrical properties. In this review, we present some essential ingredients in the flow paradigm, including the hydrodynamic modeling, the characterization of initial state geometry and the medium response relations. The extension of the flow paradigm to small colliding systems is also discussed.
基金supported by National Natural Science Foundation of China(Grant Nos.11025107,10831008 and 10901165)the Fundamental Research Funds for Central Universities(Grant No.201034000-3162643)+2 种基金High Level Talent Project in High Schools in Guangdong Province(Grant No.34000-5221001)the Fundamental Research Funds for the Central Universities(Grant No.101gpy25)China Post-doctoral Science Foundation(Grant No.201003382)
文摘We construct the canonical solitons,in terms of Cabezas-Rivas and Topping,associated with some generalized Ricci flows.
基金the National Natural Science Foundation of China (No.10071013).
文摘The partial regularity of the weak heat flow of harmonic maps from a Riemannian manifold Al with boundary into general compact Riemannian manifold N without boundary is consid-ered. It is shown that the singular set Sing(u) of the weak heat flow satisfies H(Sing(u)) 0, with is = dimensionM. Here is Hausdorff measure with respect to parabolic metric ρ(x,t),(y,s)=max{|x-y|, }.
