Surfaces In 3-manifolds Research Articles

On a Minkowski-like inequality for asymptotically flat static manifolds

The Minkowski inequality is a classical inequality in differential geometry giving a bound from below on the total mean curvature of a convex surface in Euclidean space, in terms of its area. Recently there has been interest in proving versions of this inequality for manifolds other than R n \mathbb {R}^n ; for example, such an inequality holds for surfaces in spatial Schwarzschild and AdS-Schwarzschild manifolds. In this note, we adapt a recent analysis of Y. Wei to prove a Minkowski-like inequality for general static asymptotically flat manifolds.

Morse theory for minimal surfaces in manifolds

A Morse theory of a given function gives information of the numbers of critical points of some topological type. A minimal surface, bounded by a given curve in a manifold, is characterized as a harmonic extension of a critical point of the functional \({\mathcal E}\) which corresponds to the Dirichlet integral. We want to obtain Morse theories for minimal surfaces in Riemannian manifolds. We first investigate the higher differentiabilities of \({\mathcal E}\). We then develop a Morse inequality for minimal surfaces of annulus type in a Riemannian manifold. Furthermore, we also construct body handle theories for minimal surfaces of annulus type as well as of disc type. Here we give a setting where the functional \({\mathcal E}\) is non-degenerated.

Entropy of embedded surfaces in quasi-Fuchsian manifolds

We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, $\mathbb{H}^3$, and on invariant disks embedded in $\mathbb{H}^3$. We give a rigidity theorem for all embedded surfaces when the action is Fuchsian and a rigidity theorem for negatively curved surfaces when the action is quasi-Fuchsian.

Unstable CMC spheres and outlying CMC spheres in AF 3-manifolds

In this paper, we introduce a nonlinear ODE method to construct constant mean curvature (CMC) surfaces in Riemannian manifolds with symmetry. As an application, we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with metrics like [Formula: see text]. The existence of unstable CMC spheres tells us that the stability condition in Qing–Tian’s work [On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds, J. Amer. Math. Soc. 20(4) (2007) 1091–1110] cannot be removed generally.

Upper Bounds for the First Stability Eigenvalue of Surfaces in 3-Riemannian Manifolds

Our target in this paper is given upper bounds for the first stability eigenvalue of closed (compact without boundary) surfaces in a 3-Riemannian manifold endowed with a smooth density function. As consequence, we deduce a topological constraint for the existence of closed stable surfaces in non-negatively curved spaces and a result of no existence of closed stable self-shrinkers of the mean curvature flow in $\mathbb {R}^{3}$ .

On biconservative surfaces

We study in a uniform manner the properties of biconservative surfaces in arbitrary Riemannian manifolds. Biconservative surfaces being characterized by the vanishing of the divergence of a symmetric tensor field S2 of type (1,1), their properties will follow from the general properties of a symmetric divergence-free tensor field of type (1,1). We find the link between biconservativity, the property of the shape operator AH to be a Codazzi tensor field, the holomorphicity of a generalized Hopf function and the quality of the surface to have constant mean curvature. Finally we determine a Simons type formula for biconservative surfaces and then use it to study their geometry.

Finding Non-orientable Surfaces in 3-Manifolds

We investigate the complexity of finding an embedded non-orientable surface of Euler genus g in a triangulated 3-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into 3-manifolds. We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.

On Willmore Legendrian surfaces in $$\mathbb {S}^5$$ S 5 and the contact stationary Legendrian Willmore surfaces

In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in Luo ( arXiv:1211.4227v6 ) to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in $$\mathbb {S}^5$$ , and then we use this relation to prove a classification result for Willmore Legendrian spheres in $$\mathbb {S}^5$$ . We also get an integral inequality for Willmore Legendrian surfaces and in particular we prove that if the square length of the second fundamental form of a Willmore Legendrian surface in $$\mathbb {S}^5$$ belongs to [0, 2], then it must be 0 and L is totally geodesic or 2 and L is a flat minimal Legendrian tori, which generalizes the result of Yamaguchi et al. (Proc Am Math Soc 54:276–280, 1976). We also study variation of the Willmore functional among Legendrian surfaces in 5-dimensional Sasakian manifolds. Let $$\Sigma $$ be a closed surface and $$(M,\alpha ,g_\alpha ,J)$$ a 5-dimensional Sasakian manifold with a contact form $$\alpha $$ , an associated metric $$g_\alpha $$ and an almost complex structure J. Assume that $$f:\Sigma \mapsto M$$ is a Legendrian immersion. Then f is called a contact stationary Legendrian Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations. To investigate the existence of csL Willmore surfaces we introduce a higher order flow which preserves the Legendre condition and decreases the Willmore energy. As a first step we prove that this flow is well posed if $$(M,\alpha ,g_\alpha ,J)$$ is a Sasakian Einstein manifold, in particular $$\mathbb {S}^5$$ .

Bounds on genus and configurations of embedded surfaces in 4-manifolds

For several embedded surfaces with zero self-intersection number in 4-manifolds, we show that an adjunction-type genus bound holds for at least one of the surfaces under certain conditions. For example, we derive certain adjunction inequalities for surfaces embedded in $m\mathbb{CP}^2\# n(-\mathbb{CP}^2)$ ($m, n \geq 2$). The proofs of these results are given by studying a family of Seiberg-Witten equations.

Remarks on inhomogeneous anisotropic cosmology

Recently a new no-global-recollapse argument was given for some inhomogeneous and anisotropic cosmologies that utilizes surface deformation by the mean curvature flow. In this paper we discuss important properties of the mean curvature flow of spacelike surfaces in Lorentzian manifolds. We show that singularities may form during cosmic evolution and the theorems forbidding the global recollapse lose their validity. The time evolution of the spatial scalar curvature that may kinematically prevent the recollapse is determined in normal coordinates, which shows the impact of inhomogeneities explicitly. Our analysis indicates a caveat in numerical solutions that give rise to inflation.

Foliated vector fields without periodic orbits

In this article parametric versions of Wilson’s plug and Kuperberg’s plug are discussed. We show that there is a weak homotopy equivalence induced by the inclusion between the space of non-singular vector fields tangent to a foliation and its subspace comprised of those without closed orbits, as long as the leaves of the foliation have dimension at least 3. We contrast this with the case of foliations by surfaces in 3-manifolds.

On geometric properties of isoperimetric surfaces in compact three-dimensional manifolds

This article mainly contains two theorems. In the first theorem, we obtain an upper bound for the area of isoperimetric surface that encloses small volume in a compact 3-manifold of which the scalar curvature is not less than the scalar curvature of corresponding space form. We also indicate that the 3-manifold must have constant sectional curvature when the upper bound is attained. In the second theorem, we assume the closed 3-manifold has nonnegative Ricci curvature, finite fundamental group and its scalar curvature is not less than $6$. Under these conditions, if the area of a cetain isoperimetric surface lies in a specific interval, then we will have an upper bound estimate for the volume of the 3-manifold.

Knotted surfaces in 4-manifolds and stabilizations

In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifold, which can moreover assumed to be attached in a standard way (locally and unknottedly) in many favorable situations. In particular, any exotically knotted pair of surfaces with cyclic fundamental group complements become smoothly isotopic after a same number of standard stabilizations - analogous to C.T.C. Wall's celebrated result on the stable equivalence of simply-connected 4-manifolds. We moreover show that all constructions of exotic knottings of surfaces we are aware of, which display a good variety of techniques and ideas, produce surfaces that become smoothly isotopic after a single stabilization.

Note on spun normal surfaces in 1-efficient ideal triangulations

This is an announcement of the results in the paper~[4] which is one of the series of papers dealing with the existence of spun normal surfaces in 3-manifolds with ideal triangulations. We give short comments for proofs of the results without details.

Splitting of 3-manifolds and rigidity of area-minimising surfaces

In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient manifold are weaker but the assumptions on the surface are considerably more restrictive. We then use our comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature.

A note on Weingarten hypersurfaces in the warped product manifold

In this paper, we consider the closed embedded hypersurface Σ in the warped product manifold [Formula: see text] equipped with the metric g = dr2 + λ(r)2 gN. We give some characterizations of slice {r} × N by the condition that Σ has constant weighted higher-order mean curvatures (λ′)αpk, or constant weighted higher-order mean curvature ratio (λ′)αpk/p1, which generalize Brendle's [Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 247–269] and Brendle–Eichmair's [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] results. In particular, we show that the assumption convex of Brendle–Eichmair's result [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] is unnecessary. Here pk is the kth normalized mean curvature of the hypersurface Σ. As a special case, we also give some characterizations of geodesic spheres in ℝn, ℍn and [Formula: see text], which generalize the classical Alexandrov-type results.

Surfaces in 4-Manifolds: Concordance, Isotopy, and Surgery

In this paper we will show that two surfaces of the same genus and homology class in a simply connected 4-manifold are concordant. We will show they are often topologically isotopic when their complements have cyclic fundamental group. Finally, we will show that if they are 0-concordant, then surgery on one is equivalent to surgery on the other.

Separability of embedded surfaces in 3-manifolds

AbstractWe prove that if$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$is a properly embedded$\pi _1$-injective surface in a compact 3-manifold$M$, then$\pi _1S$is separable in$\pi _1M$.

Bubble tree of branched conformal immersions and applications to the Willmore functional

We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in ${\Bbb R}^n$ with uniformly bounded areas and Willmore energies. The compactness property is applied to construct Willmore type surfaces in compact Riemannian manifolds. This includes (a) existence of a Willmore $2$-sphere in ${\Bbb S}^n$ with at least 2 nonremovable singular points (b) existence of minimizers of the Willmore functional with prescribed area in a compact manifold $N$ provided (i) the area is small when genus is $0$ and (ii) the area is close to that of the area minimizing surface of Schoen-Yau and Sacks-Uhlenbeck in the homotopy class of an incompressible map from a surface of positive genus to $N$ and $\pi_2(N)$ is trivial (c) existence of smooth minimizers of the Willmore functional if a Douglas type condition is satisfied.

Open book foliation

We study open book foliations on surfaces in 3-manifolds, and give applications to contact geometry of dimension 3. We prove a braid-theoretic formula of the self-linking number of transverse links, which reveals an unexpected link to the Johnson-Morita homomorphism in mapping class group theory. We also give an alternative combinatorial proof to the Bennequin-Eliashberg inequality.