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1.
A surface x: M S
n
is called a Willmore surface if it is a criticalsurface of the Willmore functional
M
(S – 2H
2)dv, where H isthe mean curvature and S is the square of the length of the secondfundamental form. It is well known that any minimal surface is aWillmore surface. The first nonminimal example of a flat Willmoresurface in higher codimension was obtained by Ejiri. This example whichcan be viewed as a tensor product immersion of S
1(1) and a particularsmall circle in S
2(1), and therefore is contained in S
5(1) gives anegative answer to a question by Weiner. In this paper we generalize theabove mentioned example by investigating Willmore surfaces in S
n
(1)which can be obtained as a tensor product immersion of two curves. We inparticular show that in this case too, one of the curves has to beS
1(1), whereas the other one is contained either in S
2(1) or in S
3(1). In the first case, we explicitly determine the immersion interms of elliptic functions, thus constructing infinetely many newnonminimal flat Willmore surfaces in S
5. Also in the latter casewe explicitly include examples. 相似文献
2.
QiaoLingXIA YiBingSHEN 《数学学报(英文版)》2004,20(6):1029-1046
In this paper, we reformulate the Euler-Lagrange equations of Willmore surfaces in S^n as the flatness of a family of certain loop algebra-valued 1-forms. Therefore we can give the Weierstrass type representation of conformal Willmore surfaces. We also discuss the relations between conformal Willmore surfaces in S^n and minimal surfaces in constant curvature spaces S^n, R^n, H^n, and prove that some special Willmore surfaces can be derived from minimal surfaces in S^n, R^n, H^n. 相似文献
3.
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
4.
Emilio Musso 《Annals of Global Analysis and Geometry》1990,8(1):21-41
Willmore immersions of an orientable surface X in the n-dimensionalsphere appear as the extremal points of a conformally invariant variational problem in the space of all immersions f: X S
n.In this paper we will study Willmore immersions of the differentiable two-sphere in S
4, using the method of moving frames and Cartan's conformal structures.The work on this paper was partially supported by a Fellowship of the Consiglio Nazionale delle Ricerche. 相似文献
5.
Tristan Rivière 《Inventiones Mathematicae》2008,174(1):1-45
A new formulation for the Euler–Lagrange equation of the Willmore functional for immersed surfaces in ℝ
m
is given as a nonlinear elliptic equation in divergence form, with non-linearities comprising only Jacobians. Letting be the mean curvature vector of the surface, our new formulation reads , where is a well-defined locally invertible self-adjoint elliptic operator. Several consequences are studied. In particular, the
long standing open problem asking for a meaning to the Willmore Euler–Lagrange equation for immersions having only L
2-bounded second fundamental form is now solved. The regularity of weak Willmore immersions with L
2-bounded second fundamental form is also established. Its proof relies on the discovery of conservation laws which are preserved
under weak convergence. A weak compactness result for Willmore surfaces with energy less than 8π (the Li–Yau condition ensuring
the surface is embedded) is proved, via a point removability result established for Wilmore surfaces in ℝ
m
, thereby extending to arbitrary codimension the main result in [KS3]. Finally, from this point-removability result, the strong
compactness of Willmore tori below the energy level 8π is proved both in dimension 3 (this had already been settled in [KS3])
and in dimension 4. 相似文献
6.
Manuel Barros 《Annals of Global Analysis and Geometry》1992,10(3):219-226
In this paper we deal with the following particular case of a weaker conjecture by B. Y. Chen: Are there 2-type Willmore surfaces in E 3? In particular we prove that the above question has a negative answer when the surface is the image under stereographic projection of a minimal surface in S 3. 相似文献
8.
In [17] the third author presented Moebius geometry for sub-manifolds in Sn and calculated the first variational formula of the Willmore functional by using Moebius invariants. In this paper we present the second variational formula for Willmore submanifolds. As an application of these variational formulas we give the standard examples of Willmore hypersurfaces $ \lbrace W_{k}^{m}:= S^{k}(\sqrt {(m-k)/m}) \times S^{m-k}(\sqrt {k/m}), 1 \leq k \leq m-1 \rbrace $ in Sm+1 (which can be obtained by exchanging radii in the Clifford tori $ S^{k}(\sqrt {k/m}) \times S^{m-k}(\sqrt {(m-k)/m)})$ and show that they are stable Willmore hypersurfaces. In case of surfaces in S3, the stability of the Clifford torus $ S^{1}{({1\over \sqrt {2}})}\times S^{1}{({1\over \sqrt {2}})} $ was proved by J. L. Weiner in [18]. We give also some examples of m-dimensional Willmore submanifolds in an n-dimensional unit sphere Sn. 相似文献
9.
Let
be an n-dimensional submanifold in an (n + p)-dimensional unit sphere S
n + p
, M is called a Willmore submanifold (see [11], [16]) if it is a critical submanifold to the Willmore functional
, where
is the square of the length of the second fundamental form, H is the mean curvature of M. In [11], the second author proved an integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in S
n + p
. In this paper, we discover that a similar integral inequality of Simons’ type still holds for the critical submanifolds
of the functional
. Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation. 相似文献
10.
Peter Krauter 《Geometriae Dedicata》1994,51(3):287-303
We give a complete list of affine minimal surfaces inA
3 with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inA
n
,n4, the second variation of the affine surface area is negative definite under certain conditions on the meridian. 相似文献
11.
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 \(\mathbb {S}^5\). 相似文献
12.
DENG Yanjuan & WANG Changping LMAM School of Mathematical Sciences Peking University Beijing China 《中国科学A辑(英文版)》2006,49(1):75-85
Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13. 相似文献
13.
In this paper we construct many examples of n-dimensionalWillmore Lagrangian submanifolds in the complex Euclidean space C
n
. We characterize them as the only Willmore Lagrangian submanifolds invariant under the action of SO(n). The mostimportant contribution of our construction is that it provides examplesof Willmore Lagrangian spheres in C
n
for all n 2. 相似文献
14.
We establish a barrier principle for minimal submanifolds of a Riemannianmanifold of arbitrary codimension. We construct examples of barriers fortwo-dimensional minimal surfaces in n, n 4, and apply these to deduceexistence as well as nonexistence theorems for Plateau's problem. 相似文献
15.
Wolfgang Kühnel 《Monatshefte für Mathematik》1979,87(2):123-131
The integrated square of the mean curvature of the standard torus (anchor-ring) in euclidean three-space is greater or equal to 22 with equality precisely for radii with the ratio
. The same lower bound holds for flat tori in euclidean four-space which are products of two circles. Here equality stands for the Clifford-tori having radii with the ratio 11. Several authors have generalized this result to a larger class of surfaces of the torus-type (Willmore, Chen, Shiohama andTakagi). In this note we consider the same situation for certain submanifolds of the type ofS
1×S
3 andS
2×S
2. We consider not only the trace of the second fundamental tensor (mean curvature) but also the second elementary function of its eigenvalues, which intrinsically is just the scalar curvature. The results differ from the case of the tori: at first the minimal ratio of radii is not always algebraic, secondly the lower bounds are not the same for hypersurfaces and products. 相似文献
16.
We consider Ribaucour transformations between minimal surfaces and we relate such transformations to generating planar embedded ends. Applying Ribaucour transformations to Enneper's surface and to the catenoid, we obtain new families of complete, minimal surfaces, of genus zero, immersed in R
3, with infinitely many embedded planar ends or with any finite number of such ends. Moreover, each surface has one or two nonplanar ends. A particular family is obtained from the catenoid, for each pair (n,m), nm, such that n
m0 is an irreducible rational number. For any such pair, we get a 1-parameter family of finite total curvature, complete minimal surfaces with n+2 ends, n embedded planar ends and two nonplanar ends of geometric index m, whose total curvature is –4(n+m). The analytic interpretation of a Ribaucour transformation as a Bäcklund type transformation and a superposition formula for the nonlinear differential equation = e-2 is included. 相似文献
17.
LetM be a compact minimal surface inS
3. Y. J. Hsu[5] proved that if S222, thenM is either the equatorial sphere or the Clifford torus, whereS is the square of the length of the second fundamental form ofM, ·2 denotes theL
2-norm onM. In this paper, we generalize Hsu's result to any compact surfaces inS
3 with constant mean curvature.Supported by NSFH. 相似文献
18.
Summary A simply branched minimal surface in 3 cannot be a non-degenerate critical point of Dirichlet's energy since the Hessian always has a kernel. However such minimal surface can be non-degenerate in another sense introduced earlier by R. Böhme and the author. Such surfaces arise as the zeros of a vector field on the space of all disc surfaces spanning a fixed contour. In this paper we show that the winding number of this vector field about such a surface is ±2
p
, wherep is the number of branch points. As a consequence we derive the Morse inequalities for disc minimal surfaces in 3, thereby completing the program initiated by Morse, Tompkins, and Courant. Finally, this result implies that certain contours in 4 arbitrarily close to the given contour must span at least 2
p
disc minimal surfaces. 相似文献
19.
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
20.
In this paper we develop the theory of Willmore sequences for Willmore surfaces in the 4-sphere. We show that under appropriate
conditions this sequence has to terminate. In this case the Willmore surface either is the Twistor projection of a holomorphic
curve into or the inversion of a minimal surface with planar ends in . These results give a unified explanation of previous work on the characterization of Willmore spheres and Willmore tori
with non-trivial normal bundles by various authors.
K. Leschke thanks the Department of Mathematics and Statistics at the University of Massachusetts, Amherst, and the Center
for Geometry, Analysis, Numerics and Graphics for their support and hospitality. 相似文献