arXiv subjects
Publications and source records attributed to Jintian Zhu.
The classical Llarull theorem states that a smooth metric on $n$-sphere cannot have scalar curvature no less than $n(n-1)$ and dominate the standard spherical metric at the same time unless it is the standard spherical metric. In this work, we prove that Llarull's rigidity theorem holds for $L^{\infty}$ metrics on spheres with finitely many points punctured. This is related to a question of Gromov.
In this paper, we are able to prove an analogy of the Calabi-Yau theorem for complete Riemannian manifolds with nonnegative scalar curvature which are aspherical at infinity. The key tool is an existence result for arbitrarily large bounded regions with weakly mean-concave boundary in Riemannian manifolds with sublinear volume growth. As an application, we use the same tool to show that a complete contractible Riemannian $3$-manifold with positive scalar curvature and sublinear volume growth is necessarily homeomorphic to $\mathbb R^3$.
In this paper, we investigate the weighted mass for weighted manifolds. By establishing a version of density theorem and generalizing Geroch conjecture in the setting of $P$-scalar curvature, we are able to prove the positive weighted mass theorem for weighted manifolds, which generalizes the result of Baldauf-Ozuch to non-spin manifolds.
Based on the $μ$-bubble method we are able to prove the following version of Riemannian Penrose inequality without horizon: if $g$ is a complete metric on $\mathbb R^3\setminus\{O\}$ with nonnegative scalar curvature, which is asymptotically flat around the infinity of $\mathbb R^3$, then the ADM mass $m$ at the infinity of $\mathbb R^3$ satisfies $m\geq \sqrt{\frac{A_g}{16π}}$, where $A_g$ is denoted to be the area infimum of embedded closed surfaces homologous to $\mathbb S^2(1)$ in $\mathbb R^3\setminus\{O\}$. Moreover, the equality holds if and only if there is a strictly outer-minimizing minimal $2$-sphere such that the region outside is isometric to the half Schwarzschild manifold with mass $\sqrt{\frac{A_g}{16π}}$.
Motivated by the recent progress on positive mass theorem for asymptotically flat manifolds with arbitrary ends and the Gromov's definition of scalar curvature lower bound for continuous metrics, we start a program on the positive mass theorem for asymptotically flat manifolds with $C^0$ arbitrary ends. In this work as the first step, we establish the positive mass theorem of asymptotically flat manifolds with $C^0$ arbitrary ends when the metric is $W^{1,p}_{\mathrm{loc}}$ for some $p\in(n,\infty]$ and is smooth away from a non-compact closed subset with Hausdorff dimension $n-\frac{p}{p-1}$. New techniques are developed to deal with non-compactness of the singular set.
Up to dimension five, we can prove that given any closed Riemannian manifold with nonnegative scalar curvature, of which the universal covering has vanishing homology group $H_k$ for all $k\geq 3$, either it is flat or it has Gauss-Bonnet quantity (defined by (1.3)) no greater than $8π$. In the second case, the equality for Gauss-Bonnet quantity yields that the universal covering splits as the Riemannian product of a $2$-sphere with non-negative sectional curvature and the Euclidean space. We also establish a dominated version of this result and its application to homotopical $2$-systole estimate unifies the results from [BBN10] and [Zhu20].
In this article, we give a proof for positive mass theorem of asymptotically flat manifolds with arbitrary ends when the dimension is no greater than seven. As an application, we also show a positive mass theorem for asymptotically locally Euclidean manifolds with necessary incompressible conditions.
Let $ Ω\subsetneq \mathbf{R}^n\,(n\geq 2)$ be an unbounded convex domain. We study the minimal surface equation in $Ω$ with boundary value given by the sum of a linear function and a bounded uniformly continuous function in $ \mathbf{R}^n$. If $ Ω$ is not a half space, we prove that the solution is unique. If $ Ω$ is a half space, we prove that graphs of all solutions form a foliation of $Ω\times\mathbf{R}$. This can be viewed as a stability type theorem for Edelen-Wang's Bernstein type theorem in \cite{EW2021}. We also establish a comparison principle for the minimal surface equation in $Ω$.
In this paper, we prove for $n\leq 7$ that if a differentiable $n$-manifold contains a relatively incompressible essential hypersurface in some class $\mathcal C_{deg}$, then it admits no complete metric with positive scalar curvature. Based on this result, we show for $n\leq 7$ that surgeries between orientable $n$-manifolds and $n$-torus along incompressible sub-torus with codimension no less than $2$ still preserve the obstruction for complete metrics with positive scalar curvature. As an application, we establish positive mass theorem with incompressible conditions for asymptotically flat/conical manifolds with flat fiber $F$ (including ALF and ALG manifolds), which can be viewed as a generalization of the classical positive mass theorem from \cite{SY79PMT} and \cite{SY2017}. Finally, we investigate Gromov's fill-in problem and bound the total mean curvature for nonnegative scalar curvature fill-ins of flat $2$-toruses (an optimal bound is obtained for product $2$-toruses). This confirms the validity of Mantoulidis-Miao's definition of generalized Brown-York mass in \cite{MM2017} for flat $2$-toruses.
Let $X$ be an $n$-dimensional Riemannian manifold with "large positive" scalar curvature. In this paper, we prove in a variety of cases that if $X$ "spreads" in $(n-2)$ directions {\it "distance-wise"}, then it {\it can't} much "spread" in the remaining 2-directions {\it "area-wise".} Here is a geometrically transparent example of what we plan prove in this regard that illustrates the idea. Let $g$ be a Riemannin metric on $X= S^2\times \mathbb R^{n-2}$, for which the submanifolds $$\mbox {$\mathbb R_s^{n-2}=s\times \mathbb R^{n-2}\subset X$ and $S^2_y= S^2\times y \subset X$}$$ are {\it mutually orthogonal} at all intersection points $$x=(s,y)\in X=\mathbb R_s^{n-2}\cap S^2_y.$$ (An instance of this is $g=g(s,y)=ϕ(s,y)^2ds^2+ψ(s,y)^2dy^2$.) Let the Riemannian metric on $\mathbb R_s^{n-2}$ induced from $(X,g)$, that is $g|_{\mathbb R_s^{n-2}}$, be {\it greater than the Euclidean} metric on $\mathbb R_s^{n-2} =\mathbb R^{n-2}$ for all $s\in S^2$. (This is interpreted as "large spread" of $g$ in the $(n-2)$ Euclidean directions.) {\sf If the {\it scalar curvature of $g$ is strictly greater than that of the unit 2-sphere}, $$Sc(g) \geq Sc(S^2)+\varepsilon=2+\varepsilon, \mbox { }\varepsilon>0,$$ then, provided $n\leq 7$, } (this, most likely, is unnecessary) {\sf there exists a smooth {\it non-contractible} spherical surface $S\subset X$, such that $$area(S)<area(S^2)=4π.$$} (This says, in a way, that $(X,g)$ "doesn't spread much area-wise" in the 2 directions complementary to the Euclidean ones.)
In this paper, we want to prove positive mass theorems for ALF and ALG manifolds with model spaces $\mathbb R^{n-1}\times \mathbb S^1$ and $\mathbb R^{n-2}\times \mathbb T^2$ respectively in dimensions no greater than $7$ (Theorem \ref{ALFPMT0}). { Different from the compatibility condition for spin structure in \cite[Theorem 2]{minerbe2008a}, we show that some type of incompressible condition for $\mathbb S^1$ and $\mathbb T^2$ is enough to guarantee the nonnegativity of the mass.} As in the asymptotically flat case, we reduce the desired positive mass theorems to those ones concerning non-existence of positive scalar curvature metrics on closed manifolds coming from generalize surgery to $n$-torus. { Finally, we investigate certain fill-in problems and obtain an optimal bound for total mean curvature of admissible fill-ins for flat product $2$-torus $\mathbb S^1(l_1)\times \mathbb S^1(l_2)$.}
By making use of the nice behavior of Hawking masses of 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 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 ADS-Schwarzschild manifolds with positive mass is bigger than or equal to the total mass, which is completely different from the situation in asymptotically flat case.
In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger-Gromoll splitting theorem we give a new proof to a rigidity result for complete manifolds with nonnegative scalar curvature admitting a proper smooth map to $T^{n-1}\times \mathbf R$ with nonzero degree. Here we introduce a trick to obtain the compactness of limit hypersurface from locally graphical convergence. Based on the same idea we also establish an optimal $2$-systole inequality for several classes of complete Riemannian manifolds with positive scalar curvature and further prove a rigidity result for the equality case.
In this paper, we give an affirmative answer to Gromov's conjecture ([3, Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of smooth functions on orientable open $3$-manifolds with uniformly positive sectional curvatures. For rigidity we show that the universal covering of the given manifold must be $\mathbf R^2\times (-c,c)$ with some doubly warped product metric if the optimal bound is attained. This gives a characterization for doubly warped product metrics with positive constant curvature. As a corollary, we also obtain a focal radius estimate for immersed toruses in $3$-spheres with positive sectional curvatures.
For $n\geq2,$ we obtain Liouville type theorems for minimal surface equations in half space $\mathbf R^n_+$ with affine Dirichlet boundary value or constant Neumann boundary value.
In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $(Σ,γ,H)$. We prove that given a metric $γ$ on $\mathbf{S}^{n-1}$ ($3\leq n\leq 7$), $(\mathbf{S}^{n-1},γ,H)$ admits no fill-in of NNSC metrics provided the prescribed mean curvature $H$ is large enough (Theorem \ref{Thm: no fillin nonnegative scalar 2}). Moreover, we prove that if $γ$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $\mathbf{S}^{n-1}$, then the much weaker condition that the total mean curvature $\int_{\mathbf S^{n-1}}H\,\mathrm dμ_γ$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (see P.\,23 in \cite{Gromov4}). In the second part of this paper, we investigate the $θ$-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.
We prove that the least area of the non-contractible immersed spheres is no more than $4π$ in any oriented compact manifold with dimension $n+2\leq 7$ which satisfies $R\geq 2$ and admits a map to $\mathbf S^2\times T^n$ with nonzero degree. We also prove a rigidity result for the equality case. This can be viewed as a generalization of the result in [2] to higher dimensions.
Let $(M,g)$ be a complete Riemannian $3$-manifold asymptotic to Schwarzschild-anti-deSitter and with scalar curvature $R \geq - 6$. Building on work of A.~Neves and G.~Tian and of the first-named author, we show that the leaves of the canonical foliation of $(M, g)$ are the unique solutions of the isoperimetric problem for their area. The assumption $R \geq -6$ is necessary. This is the first characterization result for large isoperimetric regions in the asymptotically hyperbolic setting that does not assume exact rotational symmetry at infinity.