Conformal and minimal immersions of compact surfaces into the 4-sphere

Bryant, Robert L.

doi:10.4310/jdg/1214437137

Cited by 243 publications

(292 citation statements)

References 5 publications

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“…A conformally immersed surface in S 3 also has a Gauss map which can be defined as follows: to each point x ∈ M 2 , attach the oriented 2-sphere S(x) in S 3 which has first order contact with f at f(x) and the same mean curvature vector there. The map x → S(x) is variously known as the central sphere congruence [1] or the conformal Gauss map [3] and is a Möbius invariant of f. The space of oriented 2-spheres in S 3 is naturally identified with the Lorentz 4-sphere which is a pseudo-Riemannian symmetric space. One shows that the harmonic map energy E(S) of S coincides with W (f) and further [1], [3] that f is Willmore if and only if S is harmonic.…”

Section: Introductionmentioning

confidence: 99%

“…3) Note that here, and elsewhere, we do not distinguish between a map f : M for some functions p, q (here and elsewhere, we use * to represent unknown functions that are irrelevant to our analysis). It is not difficult to check that the density pq du ∧ dv is independent of choices (both of asymptotic coordinates and lift) so that we have a well-defined functional…”

Section: Introductionmentioning

confidence: 99%

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Harmonic maps in unfashionable geometries

Burstall¹,

Hertrich-Jeromin²

2002

manuscripta mathematica

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Harmonic maps in unfashionable geometries

Burstall¹,

Hertrich-Jeromin²

2002

manuscripta mathematica

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“…R. Bryant [1] and Chern and Wolfson [5] showed that closed pseudoholomorphic minimal surfaces of arbitrary genus exist in S ; they called them superminimal.…”

Section: Introductionmentioning

confidence: 99%

Pseudoholomorphicity of closed minimal surfaces in constantly curved 4-spaces

Yau¹

1990

Proc. Amer. Math. Soc.

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Abstract.The twistor lifts of a minimal surface in a constantly curved 4-space are studied. Computing the Laplacian of the length of their differentials provides a topological condition for pseudoholomorphicity. Geometric conditions for holomorphicity of surfaces in flat spaces are found.The pseudoholomorphicity condition for minimal surfaces has been studied by various authors, e.g., Calabi [2], Chern [3], Eells and Wood [6]. For a long time, it has been known that the condition is fulfilled a priori when the surface is of genus zero and the ambient space is constantly curved. One would thus like to study the higher genus cases. Using the method of twistor lifts, and an analog of the diagonalization of the second fundamental form, Lemma 1.1, we obtain the Theorem. Let M be a closed minimal surface in a four-dimensional space of positive constant curvature k. Then M is pseudoholomorphic if \ x(TM ) \ > -2x(M), where x(M) and x(TM ) are the Euler characteristics of M and its normal bundle TM respectively. In this case, the immersion is determined up to an isometry of N by the curvature function of M. Furthermore, if k = 1 , then the area of M is an integral multiple of 2n.Note. R. Bryant [1] and Chern and Wolfson [5] showed that closed pseudoholomorphic minimal surfaces of arbitrary genus exist in S ; they called them superminimal.

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“…Austere submanifolds have been studied for example in [3,11]. A particularly simple (and in some sense trivial) example comes from equators: a sphere S p immersed in S n as an equator is totally geodesic, and hence the conormal bundle N * (S p ) is a special Lagrangian submanifold of T * (S n ) with respect to the Stenzel metric.…”

Section: Review Of Calibrated Geometriesmentioning

confidence: 99%

“…These surfaces are necessarily minimal, but the condition is in fact stronger (and overdetermined). See [3,14,21,37] and the references contained therein for more details.…”

Section: Calibrated Submanifolds For the Bryant-salamon Metricsmentioning

confidence: 99%

Calibrated Subbundles in Noncompact Manifolds of Special Holonomy

Karigiannis

Min-Oo

2005

Ann Glob Anal Geom

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Abstract. This paper is a continuation of [21]. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S n by looking at the conormal bundle of appropriate submanifolds of S n . We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey-Lawson for submanifolds in R n in their pioneering paper [19]. We also construct calibrated submanifolds in complete metrics with special holonomy G 2 and Spin(7) discovered by Bryant and Salamon [7] on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy.

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Conformal and minimal immersions of compact surfaces into the 4-sphere

Cited by 243 publications

References 5 publications

Harmonic maps in unfashionable geometries

Harmonic maps in unfashionable geometries

Pseudoholomorphicity of closed minimal surfaces in constantly curved 4-spaces

Calibrated Subbundles in Noncompact Manifolds of Special Holonomy

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