Semi-Parallel, Semi-Symmetric Immersions and Chen’s Equality

Recently, B. Y. Chen introduced a new invariant δ(n₁,n₂,…,n_k) of a Riemannian manifold and proved a basic inequality between the invariant and the extrinsic invariant if, where H is the mean curvature of an immersion Mⁿ in a real space form R^m(ε) of constant curvature ε. He pointed out that such inequality also holds for a totally real immersion in a complex space form. The immersion is called ideal (by B. Y. Chen) if it satisfies the equality case of such inequality identically. In this paper we classify ideal semi-parallel immersions in an Euclidean space if their normal bundle is flat, and prove that every ideal semi-parallel Lagrangian immersion in a complex space form is totally geodesic, moreover this result also holds for ideal semi-symmetric Lagrangian immersions in complex projective space and hyperbolic space.     

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M ^m of a Euclidean space R ^m+p is said to have harmonic mean curvature vector field if ${\Delta \vec{H}=0}$ , where ${\vec{H}}$ is the mean curvature vector field of ${M\hookrightarrow R^{m+p}}$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.     

On the conditions to control curvature tensors of Ricci flow

An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the full Riemannian curvature tensor. In this article, supposing (M ⁿ , g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that L^\fracn+22{L^\frac{n+2}{2}} norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C ⁰ bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds. Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M ⁿ  × [0, T), the curvature tensor stays uniformly bounded on M ⁿ  × [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented.     

Complete noncompact manifolds with harmonic curvature

Let (M ⁿ , g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (M ⁿ , g) is a space form if it has sufficiently small L ^n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (M ⁿ , g) with positive scalar curvature.     

L 2 Harmonic 1-Forms on Minimal Submanifolds in Spheres

We study a complete noncompact minimal submanifold M ⁿ in a sphere S ^n+p . We prove there is no nontrivial L ² harmonic 1-form and at most one nonparabolic end on M if the total curvature is bounded from above by a constant depending only on n. The rigidity theorem is a generalized version of Ni’s, Yun’s and the second author’s results on submanifolds in Euclidean spaces and Seo’s result on minimal submanifolds in hyperbolic spaces.     

Submanifolds with Flat Normal Bundle

Let M be a compact orientable submanifold immersed in a Riemannian manifold of constant curvature with flat normal bundle. This paper gives intrinsic conditions for M to be totally umbilical or a local product of several totally umbilical submanifolds. It is proved especially that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.     

Totally geodesic submanifolds of the complex quadric

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Qm:=SO(m+2)/(SO(2)×SO(m)) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Qm by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Qm are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]: The first type is constituted by manifolds isometric to CP¹×RP¹; their existence follows from the fact that Q² is (via the Segre embedding) holomorphically isometric to CP¹×CP¹. The second type consists of 2-spheres of radius which are neither complex nor totally real in Qm.     

Variational formulas of higher order mean curvatures

In this paper,we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional M2p of a submanifold M n in a general Riemannian manifold N n+m for p = 0,1,...,[n 2 ].As an example,we prove that closed complex submanifolds in complex projective spaces are critical points of the functional M2p,called relatively 2p-minimal submanifolds,for all p.At last,we discuss the relations between relatively 2p-minimal submanifolds and austere submanifolds in real space forms,as well as a special variational problem.     

Classification of pinched positive scalar curvature manifolds

Let (Mn,g), n?3, be a smooth closed Riemannian manifold with positive scalar curvature Rg. There exists a positive constant C=C(M,g) defined by mean curvature of Euclidean isometric immersions, which is a geometric invariant, such that Rg?n(n−1)C. In this paper we prove that Rg=n(n−1)C if and only if (Mn,g) is isometric to the Euclidean sphere Sn(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on Mn such that the scalar curvature satisfies the pinched condition

     

A note on rigidity of the almost Ricci soliton

The aim of this paper is to prove that a gradient almost Ricci soliton ${(M^{n}, g, \nabla f, \lambda )}$ whose Ricci tensor is Codazzi has constant sectional curvature. In particular, in the compact case, we deduce that (M ⁿ , g) is isometric to a Euclidean sphere and f is a height function. Moreover, we also classify gradient almost Ricci solitons with constant scalar curvature R provided a suitable function achieves a maximum in M ⁿ .     

Biharmonic Submanifolds with Parallel Mean Curvature in \mathbb{S}^{n}\times\mathbb{R}

We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces M ⁿ (c)×?, where M ⁿ (c) is a space form with constant sectional curvature c, and then we use it to prove a gap theorem for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces in $\mathbb{S}^{n}(c)\times\mathbb{R}$ .     

Yamabe Flow and Myers Type Theorem on Complete Manifolds

In this paper, we prove the following Myers type theorem: If (M ⁿ ,g), n≥3, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition Rc≥?Rg, where R>0 is the scalar curvature and ?>0 is a uniform constant, then M ⁿ must be compact.     

The curvature of the Sasaki metric of tangent sphere bundles

We study the sectional curvaturesK of the Sasaki metric of tangent sphere bundles over spaces of constant curvatureK(T ₁(M ⁿ, K)). We give precise bounds on the variation of the Ricci curvature and a bound on the scalar curvature ofT ₁ (M ⁿ, K) that is uniform onK. In an appendix we calculate and give lower bounds for the lengths of closed geodesics onT ₁ S ⁿ. titles.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 132–145.     

Some <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Rigidity Results for Complete Manifolds with Harmonic Curvature

Let (M ⁿ , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R m? the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m? goes to zero uniformly at infinity if for \(p\geq \frac n2\), the L ^p -norm of R m? is finite. Moreover, If R is positive, then (M ⁿ , g) is compact. As applications, we prove that (M ⁿ , g) is isometric to a spherical space form if for \(p\geq \frac n2\), R is positive and the L ^p -norm of R m? is pinched in [0, C ₁), where C ₁ is an explicit positive constant depending only on n, p, R and the Yamabe constant. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).     

关于局部对称空间中的伪脐子流形

本文讨论了局部对称完备黎曼流形中的紧致伪脐子流形,且具有平行平均山率向量场。得到了这类子流形成为全脐子流形及其余维数减小的几个拼挤定理。     

Transience and capacity of minimal submanifolds

We prove explicit lower bounds for the capacity of annular domains of minimal submanifolds P ^m in ambient Riemannian spaces N ⁿ with sectional curvatures bounded from above. We characterize the situations in which the lower bounds for the capacity are actually attained. Furthermore we apply these bounds to prove that Brownian motion defined on a complete minimal submanifold is transient when the ambient space is a negatively curved Hadamard-Cartan manifold. The proof stems directly from the capacity bounds and also covers the case of minimal submanifolds of dimension m > 2 in Euclidean spaces.     

Parallel Codazzi tensors with submanifold applications

A decomposition theorem is established for a class of closed Riemannian submanifolds immersed in a space form of the constant sectional curvature. In particular, it is shown that if M has nonnegative sectional curvature and admits a Codazzi tensor with “parallel mean curvature”, then M is locally isometric to a direct product of irreducible factors determined by the spectrum of that tensor. This decomposition is global when M is simply connected, and generalizes what is known for immersed submanifolds with parallel mean curvature vector.     

The Bakry–Emery Ricci tensor and its applications to some compactness theorems

Let (M, g) be a complete and connected Riemannian manifold of dimension n. By using the Bakry–Emery Ricci curvature tensor on M, we prove two theorems which correspond to the Myers compactness theorem.     

Nonexistence of complete conformal metrics with prescribed scalar curvatures on a manifold of nonpositive curvature

Let (M,g) be a complete simply-connected Riemannian manifold with nonpositive curvature, k its scalar curvature, and K a smooth function on M. We obtain a nonexistence result of complete metrics on M conformal to g and with K as their scalar curvature.     

The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu ^{(n+2)/(n–2)} on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.