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1.
A characterization of the Ejiri torus in <Emphasis Type="Italic">S</Emphasis><Superscript>5</Superscript> 下载免费PDF全文
Peng Wang 《数学学报(英文版)》2016,32(9):1014-1026
We conjecture that a Willmore torus having Willmore functional between 2π 2 and 2π 2 \(\sqrt 3 \) is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri’s torus in S 5 is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S n by reducing them into elastic curves in S 3, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S 5 attains the minimum 2π 2 \(\sqrt 3 \), which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S 5. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S 3. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough. 相似文献
2.
The sphere S
n+1 contains a simple family of constant mean curvature (CMC) hypersurfaces of the form C
t
: = S
p
(cos t) × S
q
(sin t) for p + q = n and called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces
resembling a pair of neighbouring generalized Clifford tori connected to each other by small catenoidal bridges at a sufficiently
symmetric configuration of points can be constructed by perturbative PDE methods. That is, one can create an approximate solution
by gluing a rescaled catenoid into the neighbourhood of each point; and then one can show that a perturbation of this approximate
hypersurface exists, which satisfies the CMC condition. The results of this paper generalize those of the authors in [3]. 相似文献
3.
We study some minimization problems for Hamiltonian stationaryLagrangian surfaces in R4. We show that the flat Lagrangian torusS
1 × S
1 minimizes the Willmore functional among Hamiltonianstationary tori of its isotopy class, which gives a new proof of thefact that it is area minimizing in the same class. Considering theLagrangian flat cylinder as a torus in some quotient space R4/v Z, we show that it is also area minimizing in its isotopy class. 相似文献
4.
ZhangJianfeng 《高校应用数学学报(英文版)》2005,20(2):183-196
Let M^n be a closed spacelike submanifold isometrically immersed in de Sitter space Sp^(n p)(c), Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of M^n ,respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M^n immersed in Sp^(n p)(c) with parallel normalized mean curvature vector field is proved. When n≥3, the pinching constant is the best. Thus, the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math, 1998,95 :499-505) is corrected. Moreover,the reduction of the codimension when M^n is a complete submanifold in Sp^(n p)(c) with parallel normalized mean curvature vector field is investigated. 相似文献
5.
Mark Haskins 《Inventiones Mathematicae》2004,157(1):11-70
We prove a number of results on the geometric complexity of special Lagrangian (SLG) T2-cones in 3. Every SLG T2-cone has a fundamental integer invariant, its spectral curve genus. We prove that the spectral curve genus of an SLG T2-cone gives a lower bound for its geometric complexity, i.e. the area, the stability index and the Legendrian index of any SLG T2-cone are all bounded below by explicit linearly growing functions of the spectral curve genus. We prove that the cone on the Clifford torus (which has spectral curve genus zero) in S5 is the unique SLG T2-cone with the smallest possible Legendrian index and hence that it is the unique stable SLG T2-cone. This leads to a classification of all rigid index 1 SLG cone types in dimension three. For cones with spectral curve genus two we give refined lower bounds for the area, the Legendrian index and the stability index. One consequence of these bounds is that there exist S1-invariant SLG torus cones of arbitrarily large area, Legendrian and stability indices. We explain some consequences of our results for the programme (due to Joyce) to understand the most common three-dimensional isolated singularities of generic families of SLG submanifolds in almost Calabi-Yau manifolds. Mathematics Subject Classification (1991) 53C38, 53C43 相似文献
6.
Harish Seshadri 《Mathematische Zeitschrift》2004,247(3):487-503
We study closed Einstein 4-manifolds which admit S1 actions of a certain type, i.e., warped products. In particular, we classify them up to isometry when the fixed point of the S1 action satisfies certain natural geometric conditions. The proof uses the Bochner-Weitzenböck formula for 1-forms and the theory of minimal surfaces in 3-manifolds.in final form: 22 January 2003 相似文献
7.
Xing Xiao LI Feng Yun ZHANG 《数学学报(英文版)》2007,23(3):533-548
For an immersed submanifold x : M^m→ Sn in the unit sphere S^n without umbilics, an eigenvalue of the Blaschke tensor of x is called a Blaschke eigenvalue of x. It is interesting to determine all hypersurfaces in Sn with constant Blaschke eigenvalues. In this paper, we are able to classify all immersed hypersurfaces in S^m+1 with vanishing MSbius form and constant Blaschke eigenvalues, in case (1) x has exact two distinct Blaschke eigenvalues, or (2) m = 3. With these classifications, some interesting examples are also presented. 相似文献
8.
In the paper referred to in the title, we proved that any immersed flat tori in \(S^3\) whose mean curvature does not change sign has extrinsic diameter \(\pi \) . Although the main result there is correct, there is a gap of the proof of this fact. The purpose here is to correct the previous paper’s argument and clarify the statement. 相似文献
9.
Christian Wegner 《manuscripta mathematica》2009,128(4):469-481
Let X be a finite aspherical CW-complex whose fundamental group π
1(X) possesses a subnormal series with a non-trivial elementary amenable group G
0. We investigate the L
2-invariants of the universal covering of such a CW-complex X. The main result is the proof of the vanishing of the L
2-torsion under the condition that π
1(X) has semi-integral determinant. We further show that the Novikov–Shubin invariants are positive. 相似文献
10.
In the last years there has been some interest in studying mappings in the fractional Sobolev space W1/2(, S1), see e.g., [4] [3] [15] [12] and the paper [5]. Motivated by these papers, we characterize here in the framework of Cartesian currents, see [9], the class of weak limits of sequences of smooth mappings with values into S1 with equibounded W1/2 energies. 相似文献
11.
N. A. Tyurin 《Theoretical and Mathematical Physics》2011,167(2):567-576
We propose a construction of a Lagrangian torus fibration of the full flag variety in ℂ
3
. In contrast to the classical fibration obtained from the Gelfand-Zeitlin system, the proposed fibration is special Lagrangian. 相似文献
12.
Oscar Mario Perdomo 《Geometriae Dedicata》2007,129(1):23-34
In this paper we prove that if is a minimal immersion of a compact surface and , for some homogeneous polynomial f of degree 3 on R
4, then, M is a torus and is one of the examples given by Lawson (1970, Complete minimal surfaces in S
3. Ann. Math. 92(2), 335–374).
相似文献
13.
Theodoros Vlachos 《Archiv der Mathematik》2005,85(2):175-182
We provide a characterization of the Clifford torus via a Ricci type condition among minimal surfaces in S4. More precisely, we prove that a compact minimal surface in S4, with induced metric ds2 and Gaussian curvature K, for which the metric
is flat away from points where K = 1, is the Clifford torus, provided that m is an integer with m > 2.Received: 8 September 2004 相似文献
14.
Aldo Portela 《Bulletin of the Brazilian Mathematical Society》2007,38(4):623-633
Although every Cantor subset of the circle (S1) is the minimal set of some homeomorphism of S1, not every such set is minimal for a C1 diffeomorphism of S1. In this work, we construct new examples of Cantor sets in S1 that are not minimal for any C1-diffeomorphim of S1. 相似文献
15.
Patrick LaVictoire 《Journal d'Analyse Mathématique》2011,113(1):241-263
We present a modified version of Buczolich and Mauldin’s proof that the sequence of square numbers is universally L
1-bad. We extend this result to a large class of sequences, including the dth powers and the set of primes. Furthermore, we show that any subsequence of the averages taken along these sequences is
also universally L
1-bad. 相似文献
16.
We show that the set of C
metrics in the two dimensional torus with no continuous invariant graphs of the geodesic flow is open and dense in the C
1 topology. The generic nonexistence of invariant graphs with rational rotation numbers was known in the C
topology for metrics, and in general the generic nonexistence in the C
topology of invariant graphs with Liouville rotation numbers is known for twist maps and Hamiltonian flows in the torus. The main idea of the proof is that small C
1 bumps are enough to prevent the existence of invariant graphs.Partially supported by CNPq, FAPERJ, TWAS 相似文献
17.
In this paper, we shall prove that Axiom A maps are dense in the space of C2 interval maps (endowed with the C2 topology). As a step of the proof, we shall prove real and complex a priori bounds for (first return maps to certain small neighborhoods of the critical points of) real analytic multimodal interval maps with non-degenerate critical points. We shall also discuss rigidity for interval maps without large bounds. Mathematics Subject Classification (2000) Primary 37E05; Secondary 37F25 相似文献
18.
Regina Rotman 《Mathematische Zeitschrift》2007,257(2):427-437
Let M
n
be a closed 2-connected Riemannian manifold, such that π3(M
n
) ≠ { 0 }. In this paper we prove that either there exists a periodic geodesic on M
n
of length ≤ 6d, where d is the diameter of M
n
, or at each point p ∈ M
n
there exists a geodesic loop of length ≤ 2d. 相似文献
19.
Adrian Butscher 《Annals of Global Analysis and Geometry》2009,36(3):221-274
Four constructions of constant mean curvature (CMC) hypersurfaces in
\mathbb Sn+1{\mathbb {S}^{n+1}} are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean
space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an
equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists
of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused
together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same
mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached
to each other by symmetrically positioned Delaunay-like ‘arms’. This last result extends Butscher and Pacard’s doubling construction
for generalized Clifford tori of small mean curvature. 相似文献
20.
In this paper we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T
2 for a given metric, and prove that the Hamiltonian flow on any compact level hypersurface has zero topological entropy. Furthermore,
by examples we show that the integrable Hamiltonian systems on T
2 can have complicated dynamical phenomena. For instance they can have several families of invariant tori, each family is bounded
by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders. As we know, it is the first concrete example to
present the families of invariant tori at the same time appearing in such a complicated way.
This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 10671123, 10231020), “Dawn”
Program of Shanghai Education Comission of China (Grant No. 03SG10) and Program for New Century Excellent Tatents in University
of China (Grant No. 050391) 相似文献