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Minimal surfaces in R^3 and Maximal surfaces in L^3 (Lecture 2) by Pradip Kumar

395 views · Published 28 August 2018 · 1:12:44 · Indexed 27 September 2026

Channel: International Centre for Theoretical Sciences · 2018 · Science & Technology

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ORGANIZERS : C. S. Aravinda and Rukmini Dey

DATE & TIME : 16 June 2018 to 25 June 2018

VENUE : Madhava Lecture Hall, ICTS, Bangalore


This workshop on geometry and topology for lecturers is aimed for participants who are lecturers in universities/institutes and colleges in India. This will enable lecturers from institutions all over India to get acquainted with various areas of geometry and topology which are active fields of research.

Topics will include: Geometry and Topology of surfaces; Hyperbolic geometry and moduli spaces of Riemann surfaces; Minimal surfaces, maximal surfaces and solitons; Fundamental Groupoid.

The workshop will have 4 pedagogical lecture series. There will also be discussion sessions aimed at promoting research and collaborations amongst the participants. The lecturer-participants themselves will also have a chance to present papers (which could be their own too in the area). In addition, there will be small projects which may later on develop into research projects. 


PROGRAM LINK
https://www.icts.res.in/discussion-meeting/gtl2018

Table of Contents (powered by https://videoken.com)
0:00:00 Geometry and Topology for Lecturers
0:00:10 Minimal surfaces in R caret 3 and Maximal surfaces in L caret 3 (Lecture 2)
0:00:15 Minimal and Maximal surfaces
0:00:47 Isothermal parameter 
0:06:46 Definition of minimal surface 
0:14:22 A maximal surface in the 3-dimensional Lorentz-Miukowski space E13 is a space like immersion such that its mean curvature vanishes everywhere.
0:15:27 3 dimensional Lorentz-Minkowski space E13 is a vector space R3 with the metric u. r: = u1r1 +u2r2-u3r3: u= (u1.u2.u3).r = (r1.r2.r3).
0:16:42 A vector r belongs to E13 is said to be spacelike if r. r0 or r=0. timelike if r. r0 and lightlike if r. r= 0 and r not equal to 0.
0:20:30 Let U be a subspace of R3 if induced Miukowski metric on U is positive definite we say U a spacelike subspace 
0:28:18 A (smooth) curve alpha in E13 is said to be spacelike (resp. timelike or lightlike) at t if alpha' (t) is spacelike (resp. timelike or lightlike) vector.
0:32:31 Spacelike or timelike condition is an open property.
0:36:30 Spacelike and timelike surfaces in E13
0:39:16 Let M be a compact surface and x : M - E13 be a spacelike. timelike immersion then delta M not equal to 0
0:40:04 Immersion 
0:46:46 Immersion surface in E13 may not be classified in one of the above types S2 = {{(x. y. :) belongs to R3 : x2 + y2 + - =1}
0:56:49 A hyperboloid of center p0 belongs to E13 and radius r0 is the surface H2(r. p0) = {p belongs to E13: p-p0. p - p0= -r2. p-p0. E30}
0:58:04 The Pseudosphere of center P0 and radius r is the surface S12 (r. p0) = {p belongs to E13: p-p0.p - p0= r2}
1:01:19 A maximal surface in the 3-dimensional Lorentz-Minkowski space  E13 is a spacelike immersion of 2 dimensional manifold M.
1:08:19 Let be a simply connected domain and choose three homomorphic 1-form 01. 02.03 defined on U satisfying 
1:09:22 Generalized Maximal Surface 
1:10:36 Maximal Enneper Surface.
1:11:06 Maximal Helicoid Surface

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