Difference between revisions of "User:Camillo.delellis/sandbox"
| Line 28: | Line 28: | ||
===De Giorgi's $\varepsilon$-regularity theorem=== | ===De Giorgi's $\varepsilon$-regularity theorem=== | ||
===Simons' inequality and solution of the Bernstein's problem=== | ===Simons' inequality and solution of the Bernstein's problem=== | ||
| + | ====Stable surfaces==== | ||
===Federer's estimate of the singular set=== | ===Federer's estimate of the singular set=== | ||
===Simon's rectifiability theorem=== | ===Simon's rectifiability theorem=== | ||
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==Pitts' theory== | ==Pitts' theory== | ||
===Schoen-Simon curvature estimates=== | ===Schoen-Simon curvature estimates=== | ||
| + | ===The Willmore conjecture=== | ||
==Smith's theorem and generalizations== | ==Smith's theorem and generalizations== | ||
===Applications to topology=== | ===Applications to topology=== | ||
| − | |||
=Uniqueness of tangent cones= | =Uniqueness of tangent cones= | ||
==White's theorem== | ==White's theorem== | ||
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==Taylor's theorem== | ==Taylor's theorem== | ||
==Double-bubble conjecture== | ==Double-bubble conjecture== | ||
| + | =Notable applications= | ||
=References= | =References= | ||
Revision as of 20:27, 17 October 2012
2020 Mathematics Subject Classification: Primary: 49Q15 Secondary: 49Q2049Q0528A7532C3058A2558C35 [MSN][ZBL]
An area of analysis concerned with solving geometric problems via measure-theoretic techniques. The canonical motivating physical problem is probably that investigated experimentally by J. Plateau in the nineteenth century: Given a boundary wire, how does one find the (minimal) soap film which spans it? Slightly more mathematically: Given a boundary curve, find the surface of minimal area spanning it. (Cf. also Plateau problem.) The many different approaches to solving this problem have found utility in most areas of modern mathematics and geometric measure theory is no exception: techniques and ideas from geometric measure theory have been found useful in the study of partial differential equations, the calculus of variations, geometric analysis, harmonic analysis, and fractals.
History
Measure theoretic concepts
Caratheodory construction
Hausdorff measures
Fractals
Rectifiable sets
Besicovitch's works
One-dimensional sets
General dimension and codimension
Besicovitch-Federer projection theorem
Marstrand's theorem
Besicovitch-Preiss theorem
Tangent measures
Caccioppoli sets
Functions of bounded variation
Plateau's problem in codimension 1
Existence
Regularity theory
Bernstein's problem
Simons' cone
De Giorgi's $\varepsilon$-regularity theorem
Simons' inequality and solution of the Bernstein's problem
Stable surfaces
Federer's estimate of the singular set
Simon's rectifiability theorem
Mumford Shah conjecture
Currents
Federer-Fleming theory
Compactness for integral currents
Deformation theorem
Plateau's problem in any codimension
Regularity theory
Almgren's $\varepsilon$-regularity theorem
Almgren's big regularity paper
Currents in metric spaces
Varifolds
General theory
Rectifiable and integral varifolds
Regularity theory
Allard's rectifiability theorem
Allard's $\varepsilon$-regularity theorem
Calculus of variations in the large
Pitts' theory
Schoen-Simon curvature estimates
The Willmore conjecture
Smith's theorem and generalizations
Applications to topology
Uniqueness of tangent cones
White's theorem
Simon's theorem
Lojasievicz inequality
Soap films
Almgren's $\varepsilon-\delta$ minimal sets
Taylor's theorem
Double-bubble conjecture
Notable applications
References
Camillo.delellis/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Camillo.delellis/sandbox&oldid=28497