Minimal Surfaces, Differential Geometry, and Partial Differential Equations

In summary, the conversation discussed the use of partial differential equations in observing events on minimal surfaces, as well as the connection between differential geometry and minimal surfaces from a "diffy G" perspective. The conversation also touched on the potential overlap between these studies in their use of minimal surfaces, PDEs, and "diffy G."
  • #1
octohydra
7
0
Last night in a lecture my professor explained that some partial differential equations are used to observe events on minimal surface (e.g. membranes).

A former advisor, someone that studied differential geometry, gave a brief summary of minimal surfaces but in a diffy G perspective.

1.) Are there any connections between the two studies and their perspectives on minimal surfaces?
2.) Are there any papers that use minimal surfaces, PDE's, and Diffy G together in some manner?

I'm curious to know what do PDE's and Diffy G have in 'common' via minimal surfaces. Any input is appreciated.
 
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  • #2
Minimal surfaces are the critical points of the area functional ##A(x)=\int \sqrt{g(u)}\,d^nu## with ##g(u)=(\det g_{ij}(u))## and ##g_{ij}(u)= \left( \dfrac{\partial x}{\partial u_i} \right)^\tau \dfrac{\partial x}{\partial u_j}\,.##
 

1. What are minimal surfaces?

Minimal surfaces are surfaces in three-dimensional space that have the property of minimizing surface area. This means that for a given boundary, a minimal surface will have the smallest possible surface area compared to any other surface with the same boundary. They can be described as being "flat" or "curved" in different ways, and have many interesting properties and applications in mathematics and physics.

2. What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces in different dimensions. It uses tools from calculus and linear algebra to understand the geometry of these objects, and has many applications in fields such as physics, engineering, and computer graphics.

3. How are minimal surfaces related to differential geometry?

Minimal surfaces are a major subject of study in differential geometry. The concept of minimizing surface area can be described mathematically using differential equations, and many techniques from differential geometry are used to study and classify different types of minimal surfaces.

4. What are partial differential equations?

Partial differential equations (PDEs) are mathematical equations that involve multiple variables and their partial derivatives. They are used to describe physical systems and phenomena, and are an important tool for modeling and understanding many different areas of science and engineering.

5. How are partial differential equations used to study minimal surfaces?

Partial differential equations are used to describe the behavior of minimal surfaces and to find solutions that satisfy the condition of minimizing surface area. Different types of PDEs can be used to study different types of minimal surfaces, and techniques from PDE theory are used to analyze and classify these surfaces.

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