We study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions o...We study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations with negative cosmological constant.For a static vacuum(Mn,g,V),we also compute the asymptotic expansions of g and V at conformal infinity.展开更多
By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow ...By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of the Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected C^(2)-smooth surface as initial data in asymptotically anti-de Sitter-Schwarzschild manifolds with positive mass is greater than or equal to the total mass, which is completely different from the situation in the asymptotically flat case.展开更多
In this paper,we consider the problem of the nonnegative scalar curvature(NNSC)-cobordism of Bartnik data(∑_(1)^(n-1),γ_(1),H_(1))and(∑_(2)^(n-1),γ_(2),H_(2)).We prove that given two metricsγ_(1)andγ_(2)on S^(n-...In this paper,we consider the problem of the nonnegative scalar curvature(NNSC)-cobordism of Bartnik data(∑_(1)^(n-1),γ_(1),H_(1))and(∑_(2)^(n-1),γ_(2),H_(2)).We prove that given two metricsγ_(1)andγ_(2)on S^(n-1)(3≤n≤7)with H_(1)fixed,then(S^(n-1),γ_(1),H_(1))and(S^(n-1),γ_(2),H_(2))admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough(see Theorem 1.3).Moreover,we show that for n=3,a much weaker condition that the total mean curvature∫_(s^(2))H_(2)dpγ_(2)is large enough rules out NNSC-cobordisms(see Theorem 1.2);if we require the Gaussian curvature ofγ_(2)to be positive,we get a criterion for nonexistence of the trivial NNSCcobordism by using the Hawking mass and the Brown-York mass(see Theorem 1.1).For the general topology case,we prove that(∑_(1)^(n-1),γ_(1),0)and(∑_(2)^(n-1),γ_(2),H_(2))admit no NNSC-cobordism provided the prescribed mean curvature H_(2)is large enough(see Theorem 1.5).展开更多
基金supported by National Natural Science Foundation of China (Grant Nos.10725101 and 10990013)
文摘We study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations with negative cosmological constant.For a static vacuum(Mn,g,V),we also compute the asymptotic expansions of g and V at conformal infinity.
基金supported by National Natural Science Foundation of China(Grant Nos.11671015 and 11731001)。
文摘By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of the Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected C^(2)-smooth surface as initial data in asymptotically anti-de Sitter-Schwarzschild manifolds with positive mass is greater than or equal to the total mass, which is completely different from the situation in the asymptotically flat case.
基金supported by National Natural Science Foundation of China(National Key R&D Program of China)(Grant No.11731001)Postdoctoral Science Foundation of China(Grant No.2020M680171)。
文摘In this paper,we consider the problem of the nonnegative scalar curvature(NNSC)-cobordism of Bartnik data(∑_(1)^(n-1),γ_(1),H_(1))and(∑_(2)^(n-1),γ_(2),H_(2)).We prove that given two metricsγ_(1)andγ_(2)on S^(n-1)(3≤n≤7)with H_(1)fixed,then(S^(n-1),γ_(1),H_(1))and(S^(n-1),γ_(2),H_(2))admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough(see Theorem 1.3).Moreover,we show that for n=3,a much weaker condition that the total mean curvature∫_(s^(2))H_(2)dpγ_(2)is large enough rules out NNSC-cobordisms(see Theorem 1.2);if we require the Gaussian curvature ofγ_(2)to be positive,we get a criterion for nonexistence of the trivial NNSCcobordism by using the Hawking mass and the Brown-York mass(see Theorem 1.1).For the general topology case,we prove that(∑_(1)^(n-1),γ_(1),0)and(∑_(2)^(n-1),γ_(2),H_(2))admit no NNSC-cobordism provided the prescribed mean curvature H_(2)is large enough(see Theorem 1.5).
