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Koike Lab
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A lab that centers on differential geometry and studies the time evolution of submanifolds.
AboutAbout the lab
Exploring the shapes and changes of figures inside symmetric spaces
The Koike Lab studies geometry as a whole, centering on differential geometry and including topology and geometric analysis.
Differential geometry is closely connected to Einstein's general theory of relativity, and it also relates to other areas of mathematics such as analysis and algebra.
The lab mainly investigates submanifolds in symmetric spaces and mean curvature flow, which is their time evolution.
According to the lab's website, mean curvature flow is closely related to the Ricci flow, which was used to solve the Poincaré conjecture.
Key terms
- Mean curvature flow
- A flow that deforms a submanifold in the direction of its mean curvature vector. The boundary curve of an ellipse in the plane becomes round and collapses to a single point
- Submanifold
- A figure obtained by immersing a manifold in a space of higher dimension
- Ricci flow
- A deformation that evolves a Riemannian metric as a solution of a partial differential equation called the Ricci flow equation. It was used in the solution of the Poincaré conjecture
NewsLab news
View More ↗ResearchResearch themes
The Koike Lab studies geometry as a whole, including topology and geometric analysis, centering on differential geometry, which is closely connected to Einstein's general theory of relativity. Within this, it mainly studies the theory of submanifolds in symmetric spaces.
*Illustrative imageSubmanifolds of symmetric spaces of compact type
The lab studies submanifolds in symmetric spaces of compact type by translating the problem, through a Riemannian submersion from an infinite-dimensional Hilbert space, into the study of submanifolds in the Hilbert space. A symmetric space is curved, whereas a Hilbert space is flat.
*Illustrative imageSubmanifolds of symmetric spaces of noncompact type
The lab studies submanifolds in symmetric spaces of noncompact type by translating the problem, through complexification, into the study of submanifolds in an infinite-dimensional anti-Kähler space. This removes the curvature of the space, and its advantage is that geometric structures that vanish beyond the ideal boundary appear in the complexified space.
*Illustrative imageMean curvature flow in symmetric spaces
The lab deforms submanifolds in symmetric spaces so that they satisfy the mean curvature flow equation or the inverse mean curvature flow equation, and studies how they collapse or diverge. It also modifies the equations appropriately and studies what kind of well-behaved submanifold the deformed submanifolds converge to.
*Illustrative imageMean curvature flow with surgery and the topology of hypersurfaces
The lab uses mean curvature flow with surgery, taking a certain kind of hypersurface as the initial data, to study the topology of that hypersurface. Defining mean curvature flow with surgery is extremely difficult, and research is currently not progressing easily.
MembersFaculty
- Naoyuki Koike
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Gyroid surface with Gaussian curvature” 作者不詳 / Public domain (cropped)
- Submanifolds of symmetric spaces of compact type: “Green Sphere illustration” Aizenr / CC BY-SA 4.0 (cropped)
- Submanifolds of symmetric spaces of noncompact type: “Open universe” NASA / WMAP Science Team / Public domain (cropped)
- Mean curvature flow in symmetric spaces: “Curvature comb” Carelsbergh Stijn / CC BY-SA 4.0 (cropped)
- Mean curvature flow with surgery and the topology of hypersurfaces: “Soap Films” Ryan Somma / CC BY-SA 2.0 (cropped)
- Related lab (大山口研究室): “Good luck knot(ABOK 2437)” Zaripov Rustem / CC BY-SA 4.0 (cropped)
- Related lab (山川研究室): “Algebraic surfaces” Ashton Bradley Aspir8 / CC BY-SA 4.0 (cropped)
- Related lab (佐古研究室): “Extended universe logarithmic illustration (English annotated)” Pablo Carlos Budassi / CC BY-SA 4.0 (cropped)
- Related lab (新田研究室): “First eigenspace of the Bolza surface” Joroco246 / CC BY-SA 4.0 (cropped)
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