Super-conformal surfaces associated with null complex holomorphic curves
|
|
- Derrick Potter
- 5 years ago
- Views:
Transcription
1 Super-conformal surfaces associated with null complex holomorphic curves Katsuhiro Moriya June 26, 2007 Abstract We define a correspondence from a null complex holomorphic curve in four-dimensional complex Euclidean space to a super-conformal surface in four-dimensional Euclidean space by the quaternionic theory of surfaces. As an application, we define a transformation of superconformal surfaces. 1 Introduction Construction of surfaces is an important and an interesting problem. A transformation of a surface is a way to construct a new surface from given one. Integrable system method is powerful to define transformations of surfaces. Theory of quaternionic holomorphic vector bundles [2], [3], and [6] is influenced by the theory of integrable systems and powerful to define transformations of surfaces, too. Burstall, Ferus, Leschke, Pedit, and Pinkall [2] defined the Bäcklund transforms and the Darboux transforms for Willmore surfaces in the four-dimensional sphere in terms of quaternionic holomorphic vector bundles. Generalizing it, Bohle [1] defined the Bäcklund transforms for conformal maps from a Riemann surface into the four-dimensional sphere and Leschke and Pedit [4] defined the Bäcklund transformations for quaternionic holomorphic curves. Partly supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan. 1
2 It is showed in [2] that a super-conformal surface in four-dimensional Euclidean space R 4 is a Euclidean realization of a twistor projection of a complex holomorphic curve in three-dimensional complex projective space CP 3. An arbitrary choice of two Euclidean realization define a pair of super-conformal surfaces. Then we can consider one super-conformal surface as a transformation of another super-conformal surface. Since the other relation between a complex holomorphic curve in CP 3 and a super-conformal surface except twistor projection is unclear in this correspondence, it is interesting problem to define more explicit correspondence between a complex holomorphic curve and a super-conformal surface. In this paper, we define two correspondences from a null complex holomorphic curve in four-dimensional complex Euclidean space C 4 to a superconformal surface in R 4 by the quaternionic theory of surfaces. The real part and the imaginary part of a null complex holomorphic curve in C 4 are minimal surfaces in R 4. They have a common left normal vector and a common right normal vector. Our correspondences are obtained by simple calculation using these minimal surfaces, their left normal vector, and their right normal vector. Composition of these two correspondences define a transformation between super-conformal surfaces in R 4. These correspondences are an application of a transformation we define between conformal immersions from a Riemann surface M to R 4. This transformation preserves the right normal vector of a conformal immersion. 2 Surfaces in terms of quaternions We review the quaternionic theory of surfaces [2] and [6]. The set of quaternions H is the unitary real algebra generated by the symbols i, j, and k with relations i 2 = j 2 = k 2 = 1, ij = ji = k, jk = kj = i, ki = ik = j. For a = a 0 + a 1 i + a 2 j + a 3 k H with a 0, a 1, a 2, and a 3 R, we denote by â = a 0 a 1 i a 2 j a 3 k its quaternionic conjugation. We identify H with R 4 by the identification of a 0 + a 1 i + a 2 j + a 3 k H with (a 0, a 1, a 2, a 3 ) R 4. We identify the set of complex numbers C with the unitary real algebra generated by the symbol i. Let M be a Riemann surface with its complex structure J. We denote by Ω 1 (M) the set of one-forms on M. We define an operator : Ω 1 (M) Ω 1 (M) 2
3 by ω = ω J for every ω Ω 1 (M). An immersion F : M H is said to be conformal if the induced metric on M by F from H is compatible to the complex structure of M. For an immersion F : M H, the following three conditions are equivalent (see Definition 2 and Remark 2 in [2]): 1. An immersion F is conformal. 2. There exists a unique smooth map N : M H such that (df ) = N(dF ). 3. There exists a unique smooth map R: M H such that (df ) = (df )( R). The map N is called the left normal vector of F and the map R is called the right normal vector of F. By the definition of a left normal vector and a right normal vector, we have N 2 = R 2 = 1. Hence N(dN) = (dn)n, R(dR) = (dr)r. Let F : M H be a conformal immersion with its left normal vector N and its right normal vector R. For the mean curvature vector H: M H of F, the equations 2Ĥ(dF ) = (dr) + R(dR), 2(dF )Ĥ = (dn) + N(dN) are satisfied (Proposition 8 in [2]). A conformal immersion F : M H is called a minimal surface if its mean curvature vanishes everywhere. A conformal immersion F is a minimal surface if and only if d (df ) = 0. In terms of a left normal vector N and a right normal vector R, a conformal immersion F is a minimal surface if and only if (dn) = N(dN) = (dn)n, or equivalently, (dr) = R(dR) = (dr)r. A conformal immersion F : M H is called a super-conformal surface if its curvature ellipse is a circle (see [2]). This is equivalent to that the equation (dn) = N(dN) = (dn)n 3
4 or (dr) = R(dR) = (dr)r is satisfied (see [2]). Hence the left normal vector and the right normal vector of a minimal surface are super-conformal surfaces and the left normal vector or the right normal vector of a super-conformal surface are superconformal surface around every immersed point. A super-conformal surface is a Euclidean realization of a twistor projection of a complex holomorphic curve in the three-dimensional complex projective space (Theorem 5 in [2]). Let F : M H and G: M H are conformal immersions with their left normal vector N. Then there exists a nowhere vanishing conformal immersion ψ 0 : M H such that (df ) = (dg)ψ 0 (see [6] and [4]). Then (dg)ψ 0 is called a Weierstrass representation of F in Bohle [1]. Similarly, if F and G: M H are conformal immersions with their right normal vector R, then there exists a nowhere vanishing conformal immersion ψ 1 : M H such that (df ) = ψ 1 (dg). Then ψ 1 (dg) is called a Weierstrass representation of F, too. Let F : M H be a conformal immersion with its right normal vector R F and G: M H a conformal immersion with its left normal vector N G. Then (df ) (dg) = 0 if and only if R F = N G (Proposition 16 in [2]). The map G is called a forward Bäcklund transformation of F and the map F is called a backward Bäcklund transformation of G in Bohle [1]. We see that a Bäcklund transformation of a minimal surface is a super-conformal surface and a forward Bäcklund transformation or a backward Bäcklund transformation of a super-conformal surface is a minimal surface. 3 Transformations of surfaces We make an observation on Bäcklund transformations and Weierstrass representations. We call a pair (F, G) of nowhere vanishing conformal immersions F : M H and G: M H a Bäcklund pair if (df ) (dg) = 0. Since d(f G) = (df )G + F (dg), there exists a conformal immersion from M to H with its Weierstrass representation (df )G if and only if there exists a conformal immersion from M to H with its Weierstrass representation F (dg). We say that a Bäcklund pair (F, G) is exact if there exists a conformal immersion with its Weierstrass representation (df )G, 4
5 We assume that (F 0, G 0 ) is an exact Bäcklund pair, that N F0 is the left normal vector of F 0, and that R G0 is the right normal vector of G 0. Then there exist conformal immersions F 1 : M H and G 1 : M H with their Weierstrass representations F 0 (dg 0 ) and (df 0 )G 0 respectively. We see that F 1 = F 0 G 0 G 1 and G 1 = F 0 G 0 F 1 up to constants. Indeed, d(f 0 G 0 G 1 ) = (df 0 )G 0 + F 0 (dg 0 ) (df 1 ) = (df 0 )G 0, d(f 0 G 0 F 1 ) = (df 0 )G 0 + F 0 (dg 0 ) (dg 1 ) = F 0 (dg 0 ). Because (F 0, G 0 ) is a Bäcklund pair if and only if (Ĝ0, ˆF 0 ) is, we focus on G 0 and G 1. Then G 1 is considered as a transformation of G 0 preserving the right normal vector as follows. Lemma 1. Let G 0 : M H be a nowhere vanishing conformal immersion with its left normal vector N G0 and its right normal vector R G0 We assume that there exists a nowhere vanishing conformal immersion F 0 : M H such that (F 0, G 0 ) is an exact Bäcklund pair. Let F 1 : M H be a conformal immersion such that (df 0 )G 0 is its Weierstrass representation. Then G 1 = F 0 G 0 F 1 is a conformal immersion with its left normal vector F 0 N G0 F0 1 and its right normal vector R G0. The properties of G 0 which depends on its right normal vector only are preserved under this transformation. For example, this transformation is a transformation between minimal surfaces, between super-conformal surfaces, between Lagrangian surfaces (see [5]), and between Hamiltonian-minimal Lagrangian surfaces (see [5]). The Willmore energy is preserved under these transformations (see [6]). However, this transformation may not be a transformation between Willmore surfaces since it does not transform nonconformal variations. Hence this transformation is a transformation between constrained Willmore surfaces (see [2]). 4 Null complex holomorphic curves and superconformal surfaces Applying the above lemma, we define a correspondence from a null complex holomorphic immersion in C 4 to a super-conformal surface in R 4. We consider the complexification C R H of H as C 4. Let = 2 1 (d i d) and = 2 1 (d + i d). Then a smooth map φ: M C 4 with φ = 0 is 5
6 a complex holomorphic map. For a complex holomorphic immersion φ = F 0 + if 1 with smooth maps F 0 : M H and F 1 : M H, the map F 0 is a conformal immersion with its left normal vector N and its right normal vector R if and only if the map F 1 is a conformal immersion with its left normal vector N and its right normal vector R. Indeed, an immersion φ is complex holomorphic if and only if (df 0 ) = (df 1 ). Then F 0 and F 1 are minimal surfaces since d (df 0 ) = d(df 1 ) = 0 and d (df 1 ) = d(df 0 ) = 0. Hence φ is a null complex holomorphic immersion, that is 3 m=0 ( φ m)( φ m ) = 0 for φ = φ 0 + φ 1 i + φ 2 j + φ 3 k with smooth maps φ m : M C (m = 0, 1, 2, 3). Our correspondence is defined as follows: Theorem 1. Let F 0 + if 1 : M C 4 be a nowhere vanishing null complex holomorphic immersion such that minimal surfaces F 0 : M H and F 1 : M H are nowhere vanishing and have their left normal vector N and their right normal vector R. If N and R are immersions, then G 0 = NF 0 F 1 : M H and G 1 = F 0 R F 1 : M H are super-conformal surfaces such that (dg 0 ) = N(dG 0 ) = (dg 0 )F0 1 NF 0 and that (dg 1 ) = F 0 RF0 1 (dg 1 ) = (dg 1 )R. Proof. Let F 0 + if 1 : M C 4, N : M R, and R: M R as above. Then (df 0 ) = (df 0 )( R) = (df 1 ). Hence (F 0, R) is an exact Bäcklund pair and (df 0 )R is a Weierstrass representation of F 1. Since F 0 is a minimal surface, we have (dr) = R(dR) = (dr)r. Then G 1 is a conformal immersion with its left normal vector F 0 RF0 1 and its right normal vector R by the above lemma. Since d( R) ( R)d( R) = 0, a map G 1 is a super-conformal surface. The rest of the proof is similar. Starting from a super-conformal surface for another super-conformal surface in the above theorem, we obtain a transformation of a super-conformal surface. We give a direct proof of it as follows. Corollary 1. Let G 0 : M H be a super-conformal surface with its left normal vector N G0 such that (dn G0 ) N G0 (dn G0 ) = 0. If N G0 is an immersion and a smooth map F 0 : M H defined by (dg 0 ) = (dn G0 )F 0 is a conformal immersion such that its right normal vector R F0 is an immersion, then a smooth map G 1 defined by G 1 = N G0 F 0 F 0 R F0 G 0 is a super-conformal surface with its right normal vector R F0 such that (dr F0 )+R F0 (dr F0 ) = 0. 6
7 Proof. Let G 0, N G0, F 0, and R F0 be as above. Since 0 = d(dg 0 ) = (dn G0 ) (df 0 ), the pair (N G0, F 0 ) is an exact Bäcklund pair and (df 0 ) = N G0 (df 0 ). Hence F 0 is a minimal surface and it is nowhere vanishing by its definition. Then (dg 1 ) = (dn G0 )F 0 + N G0 (df 0 ) (df 0 )R F0 F 0 (dr F0 ) (dg 0 ) = F 0 (dr F0 ). Since R F0 is an immersion, the map G 1 is a conformal immersion with its right normal vector R F0. Then G 1 is a super-conformal surface with (dr F0 ) + R F0 (dr F0 ) = 0. Defining a smooth map F 1 by F 1 = G 0 N G0 F 0, the smooth map F 0 + if 1 : M C 4 is a nowhere vanishing null complex holomorphic immersion and N G0 is the left normal vector of F 0. The right normal vector of G 0 is F0 1 N G0 F 0 and the left normal vector of G 1 is F 0 R F0 F0 1. The rigid motion of G 0 is αg 0 β +γ with quaternions α, β, and γ such that αˆα = 1 and β ˆβ = 1. Then G 1 is a rigid motion of G 0 if and only if F0 1 N G0 F 0 = βr F0 β 1 and F 0 R F0 F0 1 = αn G0 α 1. Hence G 1 is not a rigid motion of G 0 generally. References [1] C. Bohle, Möbius invariant flows of tori in S 4, Dissertation, Technischen Universität Berlin, [2] F. E. Burstall, D. Ferus, K. Leschke, F. Pedit, and U. Pinkall, Conformal geometry of surfaces in S 4 and quaternions, Lecture Notes in Mathematics, vol. 1772, Springer-Verlag, Berlin, [3] D. Ferus, K. Leschke, F. Pedit, and U. Pinkall, Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori, Invent. Math. 146 (2001), no. 3, [4] K. Leschke, and F. Pedit, Bäcklund transforms of conformal maps into the 4-sphere, PDEs, submanifolds and affine differential geometry, Banach Center Publ. 69, , Polish Acad. Sci., [5] K. Moriya, The denominators of Lagrangian surfaces in complex Euclidean plane, Mathematical Research Note , University of Tsukuba,
8 [6] F. Pedit and U. Pinkall, Quaternionic analysis on Riemann surfaces and differential geometry, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), no. Extra Vol. II, 1998, pp (electronic). Katsuhiro Moriya Institute of Mathematics, University of Tsukuba, Tennodai, Tsukuba-shi, Ibaraki-ken, Japan 8
Super-conformal surfaces associated with null complex holomorphic curves
Bull. London Math. Soc. 41 (2009) 327 331 C 2009 London Mathematical Society doi:10.1112/blms/bdp005 Super-conformal surfaces associated with null complex holomorphic curves Katsuhiro Moriya Abstract A
More informationDescription of a mean curvature sphere of a surface by quaternionic holomorphic geometry
Description of a mean curvature sphere of a surface by quaternionic holomorphic geometry Katsuhiro oriya University of Tsukuba 1 Introduction In this paper, we collect definitions and propositions from
More informationTOTALLY REAL SURFACES IN THE COMPLEX 2-SPACE
Steps in Differential Geometry, Proceedings of the Colloquium on Differential Geometry, 25 30 July, 2000, Debrecen, Hungary TOTALLY REAL SURFACES IN THE COMPLEX 2-SPACE REIKO AIYAMA Introduction Let M
More informationarxiv:dg-ga/ v1 9 Oct 1996
BONNET PAIRS AND ISOTHERMIC SURFACES GEORGE KAMBEROV, FRANZ PEDIT, AND ULRICH PINKALL arxiv:dg-ga/9610006v1 9 Oct 1996 1. Introduction A classical question in surface theory is which data are sufficient
More informationCONSTRAINED WILLMORE TORI IN THE 4 SPHERE. Christoph Bohle. Abstract
j. differential geometry 86 (2010) 71-131 CONSTRAINED WILLMORE TORI IN THE 4 SPHERE Christoph Bohle Abstract We prove that a constrained Willmore immersion of a 2 torus into the conformal 4 sphere S 4
More informationSHEAR-FREE RAY CONGRUENCES ON CURVED SPACE-TIMES. Abstract
SHEAR-FREE RAY CONGRUENCES ON CURVED SPACE-TIMES PAUL BAIRD A shear-free ray congruence (SFR) on Minkowsi space is a family of null geodesics that fill our a region of space-time, with the property that
More informationMinimal surfaces in quaternionic symmetric spaces
From: "Geometry of low-dimensional manifolds: 1", C.U.P. (1990), pp. 231--235 Minimal surfaces in quaternionic symmetric spaces F.E. BURSTALL University of Bath We describe some birational correspondences
More informationSOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE. Toshiaki Adachi* and Sadahiro Maeda
Mem. Fac. Sci. Eng. Shimane Univ. Series B: Mathematical Science 32 (1999), pp. 1 8 SOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE Toshiaki Adachi* and Sadahiro Maeda (Received December
More informationDiscrete holomorphic geometry I. Darboux transformations and spectral curves
University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Publication Series Mathematics and Statistics 2009 Discrete holomorphic geometry I. Darboux
More informationModern Geometric Structures and Fields
Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface
More informationPublished as: J. Geom. Phys. 10 (1993)
HERMITIAN STRUCTURES ON HERMITIAN SYMMETRIC SPACES F. Burstall, O. Muškarov, G. Grantcharov and J. Rawnsley Published as: J. Geom. Phys. 10 (1993) 245-249 Abstract. We show that an inner symmetric space
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationConformal maps from a 2-torus to the 4-sphere
University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Publication Series Mathematics and Statistics 2007 Conformal maps from a 2-torus to the 4-sphere
More informationSuperconformal ruled surfaces in E 4
MATHEMATICAL COMMUNICATIONS 235 Math. Commun., Vol. 14, No. 2, pp. 235-244 (2009) Superconformal ruled surfaces in E 4 Bengü (Kılıç) Bayram 1, Betül Bulca 2, Kadri Arslan 2, and Günay Öztürk 3 1 Department
More informationHYPERKÄHLER MANIFOLDS
HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly
More informationSoliton Spheres. vorgelegt von Diplom Mathematiker Günter Paul Peters aus Jena
Soliton Spheres vorgelegt von Diplom Mathematiker Günter Paul Peters aus Jena von der Fakultät II Mathematik und Naturwissenschaften der Technischen Universiät Berlin zur Erlangung des akademischen Grades
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Harmonic spinors and local deformations of the metric Bernd Ammann, Mattias Dahl, and Emmanuel Humbert Preprint Nr. 03/2010 HARMONIC SPINORS AND LOCAL DEFORMATIONS OF
More informationSelf-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds
Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: moore@math.ucsb.edu
More informationMathematische Annalen
Math. Ann. 319, 707 714 (2001) Digital Object Identifier (DOI) 10.1007/s002080100175 Mathematische Annalen A Moebius characterization of Veronese surfaces in S n Haizhong Li Changping Wang Faen Wu Received
More informationSelf-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds
Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: moore@math.ucsb.edu
More informationLegendre surfaces whose mean curvature vectors are eigenvectors of the Laplace operator
Note di Matematica 22, n. 1, 2003, 9 58. Legendre surfaces whose mean curvature vectors are eigenvectors of the Laplace operator Tooru Sasahara Department of Mathematics, Hokkaido University, Sapporo 060-0810,
More informationDoc. Math. J. DMV 313 Remarks on the Darboux Transform of Isothermic Surfaces Udo Hertrich-Jeromin 1 and Franz Pedit 2 Received: December 4, 1996 Revi
Doc. Math. J. DMV 313 Remarks on the Darboux Transform of Isothermic Surfaces Udo Hertrich-Jeromin 1 and Franz Pedit 2 Received: December 4, 1996 Revised: September 25, 1997 Communicated by Ursula Hamenstadt
More informationCONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP
CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat
More informationarxiv:math/ v1 [math.ag] 18 Oct 2003
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp. 139 152. Printed in India The Jacobian of a nonorientable Klein surface arxiv:math/0310288v1 [math.ag] 18 Oct 2003 PABLO ARÉS-GASTESI
More information7 Curvature of a connection
[under construction] 7 Curvature of a connection 7.1 Theorema Egregium Consider the derivation equations for a hypersurface in R n+1. We are mostly interested in the case n = 2, but shall start from the
More informationarxiv:math/ v1 [math.dg] 22 Dec 2000
arxiv:math/0012238v1 [math.dg] 22 Dec 2000 QUATERNIONIC HOLOMORPHIC GEOMETRY: PLÜCKER FORMULA, DIRAC EIGENVALUE ESTIMATES AND ENERGY ESTIMATES OF HARMONIC 2-TORI D. FERUS, K. LESCHKE, F. PEDIT AND U. PINKALL
More informationSYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction
SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces
More informationOn the 5-dimensional Sasaki-Einstein manifold
Proceedings of The Fourteenth International Workshop on Diff. Geom. 14(2010) 171-175 On the 5-dimensional Sasaki-Einstein manifold Byung Hak Kim Department of Applied Mathematics, Kyung Hee University,
More informationarxiv:math/ v1 [math.gr] 24 Oct 2005
arxiv:math/0510511v1 [math.gr] 24 Oct 2005 DENSE SUBSETS OF BOUNDARIES OF CAT(0) GROUPS TETSUYA HOSAKA Abstract. In this paper, we study dense subsets of boundaries of CAT(0) groups. Suppose that a group
More informationConstant Mean Curvature Tori in R 3 and S 3
Constant Mean Curvature Tori in R 3 and S 3 Emma Carberry and Martin Schmidt University of Sydney and University of Mannheim April 14, 2014 Compact constant mean curvature surfaces (soap bubbles) are critical
More information1 Introduction and preliminaries notions
Bulletin of the Transilvania University of Braşov Vol 2(51) - 2009 Series III: Mathematics, Informatics, Physics, 193-198 A NOTE ON LOCALLY CONFORMAL COMPLEX LAGRANGE SPACES Cristian IDA 1 Abstract In
More informationMinimal surfaces from self-stresses
Minimal surfaces from self-stresses Wai Yeung Lam (Wayne) Technische Universität Berlin Brown University Edinburgh, 31 May 2016 Wai Yeung Lam (Wayne) (TU Berlin) Discrete minimal surfaces 31 May 2016 1
More informationH-projective structures and their applications
1 H-projective structures and their applications David M. J. Calderbank University of Bath Based largely on: Marburg, July 2012 hamiltonian 2-forms papers with Vestislav Apostolov (UQAM), Paul Gauduchon
More informationQUATERNIONS AND ROTATIONS
QUATERNIONS AND ROTATIONS SVANTE JANSON 1. Introduction The purpose of this note is to show some well-known relations between quaternions and the Lie groups SO(3) and SO(4) (rotations in R 3 and R 4 )
More informationLie Algebra of Unit Tangent Bundle in Minkowski 3-Space
INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY VOLUME 12 NO. 1 PAGE 1 (2019) Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space Murat Bekar (Communicated by Levent Kula ) ABSTRACT In this paper, a one-to-one
More informationBIG PICARD THEOREMS FOR HOLOMORPHIC MAPPINGS INTO THE COMPLEMENT OF 2n + 1 MOVING HYPERSURFACES IN CP n
Available at: http://publications.ictp.it IC/2008/036 United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL
More informationarxiv: v1 [math.dg] 15 Aug 2011
arxiv:1108.2943v1 [math.dg] 15 Aug 2011 The Space-like Surfaces with Vanishing Conformal Form in the Conformal Space Changxiong Nie Abstract. The conformal geometry of surfaces in the conformal space Q
More informationSolvability of the Dirac Equation and geomeric applications
Solvability of the Dirac Equation and geomeric applications Qingchun Ji and Ke Zhu May 21, KAIST We study the Dirac equation by Hörmander s L 2 -method. On surfaces: Control solvability of the Dirac equation
More informationCitation Osaka Journal of Mathematics. 40(3)
Title An elementary proof of Small's form PSL(,C and an analogue for Legend Author(s Kokubu, Masatoshi; Umehara, Masaaki Citation Osaka Journal of Mathematics. 40(3 Issue 003-09 Date Text Version publisher
More informationPSEUDOHOLOMORPHICITY OF CLOSED MINIMAL SURFACES IN CONSTANTLY CURVED 4-SPACES
proceedings of the american mathematical society Volume 110, Number 4, December 1990 PSEUDOHOLOMORPHICITY OF CLOSED MINIMAL SURFACES IN CONSTANTLY CURVED 4-SPACES CHI-MING YAU (Communicated by Jonathan
More informationν(u, v) = N(u, v) G(r(u, v)) E r(u,v) 3.
5. The Gauss Curvature Beyond doubt, the notion of Gauss curvature is of paramount importance in differential geometry. Recall two lessons we have learned so far about this notion: first, the presence
More informationON HAMILTONIAN STATIONARY LAGRANGIAN SPHERES IN NON-EINSTEIN KÄHLER SURFACES
ON HAMILTONIAN STATIONARY LAGRANGIAN SPHERES IN NON-EINSTEIN KÄHLER SURFACES ILDEFONSO CASTRO, FRANCISCO TORRALBO, AND FRANCISCO URBANO Abstract. Hamiltonian stationary Lagrangian spheres in Kähler-Einstein
More informationMathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang
Mathematical Research Letters 2, 305 310 (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions
More informationWARPED PRODUCTS PETER PETERSEN
WARPED PRODUCTS PETER PETERSEN. Definitions We shall define as few concepts as possible. A tangent vector always has the local coordinate expansion v dx i (v) and a function the differential df f dxi We
More informationLecture XI: The non-kähler world
Lecture XI: The non-kähler world Jonathan Evans 2nd December 2010 Jonathan Evans () Lecture XI: The non-kähler world 2nd December 2010 1 / 21 We ve spent most of the course so far discussing examples of
More informationBubble Tree Convergence for the Harmonic Sequence of Harmonic Surfaces in CP n
Acta Mathematica Sinica, English Series Jul., 2010, Vol. 26, No. 7, pp. 1277 1286 Published online: June 15, 2010 DOI: 10.1007/s10114-010-8599-0 Http://www.ActaMath.com Acta Mathematica Sinica, English
More informationConformally flat hypersurfaces with cyclic Guichard net
Conformally flat hypersurfaces with cyclic Guichard net (Udo Hertrich-Jeromin, 12 August 2006) Joint work with Y. Suyama A geometrical Problem Classify conformally flat hypersurfaces f : M n 1 S n. Def.
More informationLECTURE 5: SURFACES IN PROJECTIVE SPACE. 1. Projective space
LECTURE 5: SURFACES IN PROJECTIVE SPACE. Projective space Definition: The n-dimensional projective space P n is the set of lines through the origin in the vector space R n+. P n may be thought of as the
More informationLECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES. 1. Introduction
LECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES 1. Introduction Until now we have been considering homogenous spaces G/H where G is a Lie group and H is a closed subgroup. The natural
More informationThe spectral curve of a quaternionic holomorphic line bundle over a 2-torus
University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Publication Series Mathematics and Statistics 2009 The spectral curve of a quaternionic holomorphic
More informationMATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, Elementary tensor calculus
MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 205 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let
More informationScalar curvature and the Thurston norm
Scalar curvature and the Thurston norm P. B. Kronheimer 1 andt.s.mrowka 2 Harvard University, CAMBRIDGE MA 02138 Massachusetts Institute of Technology, CAMBRIDGE MA 02139 1. Introduction Let Y be a closed,
More informationHyperkähler geometry lecture 3
Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843
More informationHARMONIC TORI IN QUATERNIONIC PROJECTIVE 3-SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 1, January 1997, Pages 275 285 S 0002-9939(97)03638-1 HARMONIC TORI IN QUATERNIONIC PROJECTIVE 3-SPACES SEIICHI UDAGAWA (Communicated
More informationJ-holomorphic curves in symplectic geometry
J-holomorphic curves in symplectic geometry Janko Latschev Pleinfeld, September 25 28, 2006 Since their introduction by Gromov [4] in the mid-1980 s J-holomorphic curves have been one of the most widely
More informationDIFFERENTIAL FORMS AND COHOMOLOGY
DIFFERENIAL FORMS AND COHOMOLOGY ONY PERKINS Goals 1. Differential forms We want to be able to integrate (holomorphic functions) on manifolds. Obtain a version of Stokes heorem - a generalization of the
More informationConformal Submersions of S 3
Conformal Submersions of S 3 DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Mathematik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakultät
More informationEinstein H-umbilical submanifolds with parallel mean curvatures in complex space forms
Proceedings of The Eighth International Workshop on Diff. Geom. 8(2004) 73-79 Einstein H-umbilical submanifolds with parallel mean curvatures in complex space forms Setsuo Nagai Department of Mathematics,
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More informationA NOTE ON E 8 -INTERSECTION FORMS AND CORRECTION TERMS
A NOTE ON E 8 -INTERSECTION FORMS AND CORRECTION TERMS MOTOO TANGE Abstract. In this paper we construct families of homology spheres which bound 4-manifolds with intersection forms isomorphic to E 8. We
More informationSoliton surfaces and generalized symmetries of integrable equations
Soliton surfaces and generalized symmetries of integrable equations Sarah Post, joint with Michel Grundland Centre de Recherches Mathématques Université de Montreal Symmetries in Science, Bregenz August
More informationTesi di Laurea Magistrale in Matematica presentata da. Claudia Dennetta. Symplectic Geometry. Il Relatore. Prof. Massimiliano Pontecorvo
UNIVERSITÀ DEGLI STUDI ROMA TRE FACOLTÀ DI SCIENZE M.F.N. Tesi di Laurea Magistrale in Matematica presentata da Claudia Dennetta Symplectic Geometry Relatore Prof. Massimiliano Pontecorvo Il Candidato
More informationclass # MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS
class # 34477 MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS [DG] stands for Differential Geometry at https://people.math.osu.edu/derdzinski.1/courses/851-852-notes.pdf [DFT]
More informationarxiv: v1 [math.dg] 2 Mar 2015
Geometric Properties of Conformal Transformations on R p,q Matvei Libine and Surya Raghavendran February 13, 2018 arxiv:1503.00520v1 [math.dg] 2 Mar 2015 Abstract We show that conformal transformations
More informationConstant mean curvature biharmonic surfaces
Constant mean curvature biharmonic surfaces Dorel Fetcu Gheorghe Asachi Technical University of Iaşi, Romania Brest, France, May 2017 Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May 2017 1 / 21
More informationStolz angle limit of a certain class of self-mappings of the unit disk
Available online at www.sciencedirect.com Journal of Approximation Theory 164 (2012) 815 822 www.elsevier.com/locate/jat Full length article Stolz angle limit of a certain class of self-mappings of the
More informationIV. Conformal Maps. 1. Geometric interpretation of differentiability. 2. Automorphisms of the Riemann sphere: Möbius transformations
MTH6111 Complex Analysis 2009-10 Lecture Notes c Shaun Bullett 2009 IV. Conformal Maps 1. Geometric interpretation of differentiability We saw from the definition of complex differentiability that if f
More informationCohomology of the Mumford Quotient
Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten
More informationConification of Kähler and hyper-kähler manifolds and supergr
Conification of Kähler and hyper-kähler manifolds and supergravity c-map Masaryk University, Brno, Czech Republic and Institute for Information Transmission Problems, Moscow, Russia Villasimius, September
More informationBERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS
BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate
More informationConformal foliations and CR geometry
Geometry and Analysis, Flinders University, Adelaide p. 1/16 Conformal foliations and CR geometry Michael Eastwood [joint work with Paul Baird] University of Adelaide Geometry and Analysis, Flinders University,
More information(z 0 ) = lim. = lim. = f. Similarly along a vertical line, we fix x = x 0 and vary y. Setting z = x 0 + iy, we get. = lim. = i f
. Holomorphic Harmonic Functions Basic notation. Considering C as R, with coordinates x y, z = x + iy denotes the stard complex coordinate, in the usual way. Definition.1. Let f : U C be a complex valued
More informationEta Invariant and Conformal Cobordism
Annals of Global Analysis and Geometry 27: 333 340 (2005) C 2005 Springer. 333 Eta Invariant and Conformal Cobordism XIANZHE DAI Department of Mathematics, University of California, Santa Barbara, California
More informationConformal foliations and CR geometry
Twistors, Geometry, and Physics, celebrating the 80th birthday of Sir Roger Penrose, at the Mathematical Institute, Oxford p. 1/16 Conformal foliations and CR geometry Michael Eastwood [joint work with
More informationJeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi
Bull. Korean Math. Soc. 40 (003), No. 3, pp. 411 43 B.-Y. CHEN INEQUALITIES FOR SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi Abstract. Some B.-Y.
More informationReport submitted to Prof. P. Shipman for Math 641, Spring 2012
Dynamics at the Horsetooth Volume 4, 2012. The Weierstrass-Enneper Representations Department of Mathematics Colorado State University mylak@rams.colostate.edu drpackar@rams.colostate.edu Report submitted
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday January 20, 2015 (Day 1)
Tuesday January 20, 2015 (Day 1) 1. (AG) Let C P 2 be a smooth plane curve of degree d. (a) Let K C be the canonical bundle of C. For what integer n is it the case that K C = OC (n)? (b) Prove that if
More informationThe Spinor Representation
The Spinor Representation Math G4344, Spring 2012 As we have seen, the groups Spin(n) have a representation on R n given by identifying v R n as an element of the Clifford algebra C(n) and having g Spin(n)
More informationRIGIDITY OF MINIMAL ISOMETRIC IMMERSIONS OF SPHERES INTO SPHERES. Christine M. Escher Oregon State University. September 10, 1997
RIGIDITY OF MINIMAL ISOMETRIC IMMERSIONS OF SPHERES INTO SPHERES Christine M. Escher Oregon State University September, 1997 Abstract. We show two specific uniqueness properties of a fixed minimal isometric
More informationHomogeneous para-kähler Einstein manifolds. Dmitri V. Alekseevsky
Homogeneous para-kähler Einstein manifolds Dmitri V. Alekseevsky Hamburg,14-18 July 2008 1 The talk is based on a joint work with C.Medori and A.Tomassini (Parma) See ArXiv 0806.2272, where also a survey
More informationCompact manifolds of nonnegative isotropic curvature and pure curvature tensor
Compact manifolds of nonnegative isotropic curvature and pure curvature tensor Martha Dussan and Maria Helena Noronha Abstract We show the vanishing of the Betti numbers β i (M), 2 i n 2, of compact irreducible
More informationLAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS
LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS JON WOLFSON Abstract. We show that there is an integral homology class in a Kähler-Einstein surface that can be represented by a lagrangian twosphere
More informationERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX
ERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX JOHN LOFTIN, SHING-TUNG YAU, AND ERIC ZASLOW 1. Main result The purpose of this erratum is to correct an error in the proof of the main result
More informationMetrics and Holonomy
Metrics and Holonomy Jonathan Herman The goal of this paper is to understand the following definitions of Kähler and Calabi-Yau manifolds: Definition. A Riemannian manifold is Kähler if and only if it
More informationCHARACTERISTIC CLASSES
1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact
More informationMinimal timelike surfaces in a certain homogeneous Lorentzian 3-manifold II
Minimal timelike surfaces in a certain homogeneous Lorentzian 3-manifold II Sungwook Lee Abstract The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space and
More informationThe Gauss map and second fundamental form of surfaces in R 3
The Gauss map and second fundamental form of surfaces in R 3 J. A. Gálvez A. Martínez Departamento de Geometría y Topoloía, Facultad de Ciencias, Universidad de Granada, 18071 GRANADA. SPAIN. e-mail: jaalvez@oliat.ur.es;
More informationSOME ALMOST COMPLEX STRUCTURES WITH NORDEN METRIC ON THE TANGENT BUNDLE OF A SPACE FORM
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLVI, s.i a, Matematică, 2000, f.1. SOME ALMOST COMPLEX STRUCTURES WITH NORDEN METRIC ON THE TANGENT BUNDLE OF A SPACE FORM BY N. PAPAGHIUC 1
More informationIntroduction to Minimal Surface Theory: Lecture 2
Introduction to Minimal Surface Theory: Lecture 2 Brian White July 2, 2013 (Park City) Other characterizations of 2d minimal surfaces in R 3 By a theorem of Morrey, every surface admits local isothermal
More informationWarped Products. by Peter Petersen. We shall de ne as few concepts as possible. A tangent vector always has the local coordinate expansion
Warped Products by Peter Petersen De nitions We shall de ne as few concepts as possible. A tangent vector always has the local coordinate expansion a function the di erential v = dx i (v) df = f dxi We
More informationHolomorphic line bundles
Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank
More informationLecture 7. Quaternions
Matthew T. Mason Mechanics of Manipulation Spring 2012 Today s outline Motivation Motivation have nice geometrical interpretation. have advantages in representing rotation. are cool. Even if you don t
More informationClassification problems in conformal geometry
First pedagogical talk, Geometry and Physics in Cracow, the Jagiellonian University p. 1/13 Classification problems in conformal geometry Introduction to conformal differential geometry Michael Eastwood
More information1 First and second variational formulas for area
1 First and second variational formulas for area In this chapter, we will derive the first and second variational formulas for the area of a submanifold. This will be useful in our later discussion on
More informationStable bundles on CP 3 and special holonomies
Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M
More informationarxiv: v1 [math.sg] 6 Nov 2015
A CHIANG-TYPE LAGRANGIAN IN CP ANA CANNAS DA SILVA Abstract. We analyse a simple Chiang-type lagrangian in CP which is topologically an RP but exhibits a distinguishing behaviour under reduction by one
More informationarxiv: v1 [math.dg] 17 Jan 2019
RECTANGULAR CONSTRAINED WILLMORE MINIMIZERS AND THE WILLMORE CONJECTURE arxiv:1901.05664v1 [math.dg] 17 Jan 019 LYNN HELLER, SEBASTIAN HELLER, AND CHEIKH BIRAHIM NDIAYE Abstract. We show that the well-known
More informationLECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
More informationSOME COUNTEREXAMPLES IN DYNAMICS OF RATIONAL SEMIGROUPS. 1. Introduction
SOME COUNTEREXAMPLES IN DYNAMICS OF RATIONAL SEMIGROUPS RICH STANKEWITZ, TOSHIYUKI SUGAWA, AND HIROKI SUMI Abstract. We give an example of two rational functions with non-equal Julia sets that generate
More informationarxiv:math/ v1 [math.dg] 29 Sep 1998
Unknown Book Proceedings Series Volume 00, XXXX arxiv:math/9809167v1 [math.dg] 29 Sep 1998 A sequence of connections and a characterization of Kähler manifolds Mikhail Shubin Dedicated to Mel Rothenberg
More information