12.4 Geometric measure theory and mathematical physics

Geometric measure theory bridges continuous and discrete aspects of physical systems, providing a rigorous framework for analyzing smooth and singular structures. It's crucial for modeling interfaces, phase transitions, and energy minimization problems in physics, offering tools to study complex phenomena.

Currents and varifolds, key concepts in geometric measure theory, represent physical objects with singularities and handle branching or merging surfaces. These tools are vital for studying minimal surfaces, soap films, and crystal growth, enabling analysis of existence, regularity, and stability in geometric variational problems.

Geometric Measure Theory in Physics

Rigorous Framework for Geometric and Measure-Theoretic Properties

Top images from around the web for Rigorous Framework for Geometric and Measure-Theoretic Properties

• 

  File:Mathematical Spaces.svg - Wikipedia View original

  Is this image relevant?

• 

  Manifold - Wikipedia View original

  Is this image relevant?

• 

  Manifold - Wikipedia View original

  Is this image relevant?

• 

  File:Mathematical Spaces.svg - Wikipedia View original

  Is this image relevant?

• 

  Manifold - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Rigorous Framework for Geometric and Measure-Theoretic Properties

• 

  File:Mathematical Spaces.svg - Wikipedia View original

  Is this image relevant?

• 

  Manifold - Wikipedia View original

  Is this image relevant?

• 

  Manifold - Wikipedia View original

  Is this image relevant?

• 

  File:Mathematical Spaces.svg - Wikipedia View original

  Is this image relevant?

• 

  Manifold - Wikipedia View original

  Is this image relevant?

1 of 3

• Provides a rigorous framework for studying the geometric and measure-theoretic properties of sets and functions in Euclidean spaces and on manifolds
• Enables the analysis of both smooth and singular structures (crystals, soap films)
• Bridges the continuous and discrete aspects of physical systems

Applications in Mathematical Physics

• Used to model and analyze physical phenomena involving interfaces, phase transitions, and energy minimization problems
• Applied in the study of minimal surfaces, soap films, and crystals
• Analyzes singularities in physical systems (cracks, voids, defects in materials)
• Tools of geometric measure theory, such as currents and varifolds, allow for the precise formulation and study of variational problems in physics

Currents and Varifolds for Modeling Phenomena

Currents for Representing Physical Objects

• Generalized surfaces that can be used to represent physical objects with singularities or non-smooth boundaries (cracks, voids, defects in materials)
• Extend the notion of integration to non-smooth domains
• Allow for the formulation of variational problems in a weak sense

Varifolds for Handling Singularities and Multiplicities

• Measure-theoretic generalizations of surfaces that can handle more general types of singularities and multiplicities compared to currents
• Particularly useful in modeling physical phenomena involving branching, merging, or cancellation of surfaces (soap films, crystal growth)
• Enable the analysis of the existence, regularity, and stability of minimal surfaces and other critical points of geometric variational problems in physics

Geometric Measure Theory for PDE Regularity

Tools for Studying Regularity and Singularities

• Provides tools for studying the regularity and singularities of solutions to partial differential equations (PDEs) arising in mathematical physics
• Notion of rectifiability, which characterizes sets with locally finite measure and a tangent space almost everywhere, plays a crucial role in the regularity theory of PDEs
• Techniques from geometric measure theory, such as monotonicity formulas and density estimates, are used to analyze the local behavior and singularities of solutions to PDEs

Existence and Regularity of Solutions

• Used to establish the existence and regularity of minimal surfaces and other critical points of geometric variational problems, which often arise as solutions to certain PDEs
• Theory of currents and varifolds is employed to study the regularity of weak solutions to PDEs, particularly in the context of geometric flows and free boundary problems

Geometric Measure Theory vs Gauge Theory

Gauge Theory and Geometric Measure Theory

• Gauge theory is a framework for describing the dynamics of fields and their interactions, central in quantum field theory and particle physics
• Geometric measure theory provides a natural language for formulating and studying gauge-theoretic problems, particularly in Yang-Mills theory and the geometry of connections on vector bundles

Instantons and Singularities

• Theory of currents and varifolds is used to study the existence and regularity of instantons, critical points of the Yang-Mills functional crucial in the topology of four-manifolds
• Geometric measure theory is employed to analyze the singularities and bubbling phenomena that occur in the study of gauge fields and their limiting behavior

Important Developments

• Interplay between geometric measure theory and gauge theory has led to important developments in mathematical physics
• Proof of the positive mass theorem in general relativity using the stability of the Yang-Mills-Higgs functional

Geometric Measure Theory in Relativity and Cosmology

Geometry and Topology of Spacetime

• General relativity is the geometric theory of gravitation describing the large-scale structure of spacetime and the dynamics of gravitational fields
• Geometric measure theory provides a framework for studying the geometry and topology of spacetime, particularly in the presence of singularities or non-smooth structures

Modeling Matter and Energy Distribution

• Theory of currents and varifolds is used to model and analyze the distribution of matter and energy in spacetime (cosmic strings, gravitational waves)
• Employed to study the existence and regularity of minimal surfaces and other critical points of the Einstein-Hilbert action, governing the dynamics of spacetime in general relativity

Investigating Singularities in Cosmological Models

• Tools of geometric measure theory are used to investigate the structure of singularities in cosmological models (black holes, big bang singularity)
• Study the possible resolutions or extensions of spacetime beyond these singularities