2002 Fiscal Year Final Research Report Summary
Morse index of constant mean curvature surfaces and discrete constant mean curvature surfaces
| Project/Area Number |
12640070
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | KOBE UNIVERSITY |
|---|
Principal Investigator |
ROSSMAN Wayne Kobe University Faculty of Science Associate Professor, 理学部, 助教授 (50284485)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
YAMADA Koutarou Kyushu University Grad. Sch. Math. Professor, 大学院・数理科学研究院, 教授 (10221657)
MIYAKAWA Tetsurou Nagoya University Grad. Sch. Math. Professor, 大学院・多元数理科学研究科, 教授 (10033929)
|
|---|
| Project Period (FY) |
2000 – 2002
|
| Keywords | minimal surface / constant mean curvature surfaces / Euclidean 3-space / hyperbolic 3-space / Morse index / discrete surfaces / integrable systems / spherical 3-space |
|---|
| Research Abstract |
(1) Working with Konrad Polthier, we considered a variational approach for defining discrete minimal surfaces, and established a variational approach for defining discrete constant mean curvature surfaces. We constructed discrete catenoids and helicoids and Delaunay surfaces, and we completely classified the case of catenoids. Furthermore, we computed the discrete Morse index of discrete minimal surfaces and used these computations to examine the Morse index of smooth minimal surfaces. (2) The head investigator proved that Wente tori (these are genus 1 compact constant mean curvature surfaces in R^3) have Morse index at least 7, and also found a lower bound for the Morse index of Wente tori that grows with the spectral genus of the surface. Furthermore, working with Lima and Sousa Neto, we improved the lower bound estimate of 7 to 8. (3) Working with Lima and Berard, we determined the growth rate of the Morse index on complete noncompact constant mean curvature surfaces. (4) Working with Thayer and Wohlgemuth, we constructed many examples of doubly-periodic minimal surfaces in R^3. (5) Working with Umehara and Yamada, we classified all constant mean curvature 1 surfaces in hyperbolic 3-space H^3 that have total curvature at most 8π. (6) Working with Umehara and Yamada and Kokubu, we have started a project to study the nature of singular points on flat surfaces in H^3. (7) Working with Schmitt and Kilian, we have started a project to study constant mean curvature surfaces of genus 0 with three asymptotically Delaunay ends in R^3 and H^3 and the 3-dimensional spherical space S^3.
|
