MCF-stack: Special Solutions of Mean Curvature Flow

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Special solutions of mean curvature flow

MCF-stack contributors

These contributions appear throughout the shared chapters.

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An illustrated catalogue of self-shrinkers, translators, self-expanders, and other special mean curvature flows, with formulas, geometric properties, precise references, and stated illustration conventions.

Source records: Build manifest · Reference inventory · Entry and proof status

Reference library

The library records supplied files and their study status. A catalogue entry does not mean that its results have been incorporated. Original books, scans, and extracted text are not distributed.

Cited sources and checked passages (122 files)
Catalogued for further study (86 files)
  • 1-dimensional solutions of the λ-self shrinkers · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • 2014 Schulze - Uniqueness of compact tangent flows in Mean Curvature Flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • A general regularity theory for weak mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • A local monotonicity formula for mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • A local regularity theorem for mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • A local regularity theorem for mean curvature flow with triple edges · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • A note of the Hamiltonian area conjecture · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • A Unique Continuation Property for the Level Set Equation · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Arnold-Thom Gradient Conjecture for the Arrival Time · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Asymptotic behavior for singularities of the mean curvature flowGerhard Huisken · 1990, Journal of Differential Geometry 31, pp. 285--299Primary flow reference from the local library; selected formulas and hypotheses checked for the expanded flow chapters.
  • Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Asymptotics for the level set equation near a maximum · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Brakke's mean curvature flow. An introduction · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Complexity of Parabolic System · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Convexity estimates for mean curvature flow and singularities of mean convex surfaces · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Differentiability of the Arrival Time · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Dynamic of Closed Singularities · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Elliptic regularization and partial regularity for motion by mean curvature · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Entropy and codimension bounds for generic singularities · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Evolution of Curves and Surfaces by Mean Curvature · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Flow under Curvature: Singularity Formation, Minimal Surfaces, and Geodesics · Chopp–Sethian, Experimental Mathematics 2(4), 235–255 (1993); author recompiled PDF dated August 10, 2006, with 48 PDF pagesReviewed for the Chopp catalogue addition; scope and exclusions in documentation/chopp-review.md.
  • Generalized Flow of Sets by Mean Curvature on a Manifold · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Geometric evolution equations for hypersurfacesGerhard Huisken; Alexander Polden · 1999, Lecture Notes in Mathematics 1713, pp. 45--84Primary flow reference from the local library; selected formulas and hypotheses checked for the expanded flow chapters.
  • Interior estimates for hypersurfaces moving by mean curvatureKlaus Ecker; Gerhard Huisken · Inventiones Mathematicae 105 (1991), 547–569; supplied 23-page journal copyLocal graphical angle, height-to-gradient and curvature estimates developed with proofs; existence for arbitrary locally Lipschitz entire initial graphs explicitly external. Comparison and cone-expander consequences derived locally.
  • Lectures on Mean Curvature Flow and Related Equations · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Local monotonicity and mean value formulas for evolving Riemannian manifolds · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Mean curvature evolution of entire graphsKlaus Ecker; Gerhard Huisken · Annals of Mathematics130(1989),no.3,453–471; supplied20-page journal copy including archive coverAngle and weighted-curvature evolution; graphical smoothing; global globally-Lipschitz-data existence retained as an external theorem. Periodic decay estimates and examples derived locally.
  • Mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Mean Curvature Flow of Mean Convex Hypersurfaces · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Mean curvature flow of pinched submanifolds to spheres · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Mean curvature flow singularities for mean convex surfaces · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Mean curvature flow through singularities for surfaces of rotation · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Mean curvature flow through singularities for surfaces of rotation · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Mean Curvature Flow with Boundary · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Mean curvature flow with surgeries of two–convex hypersurfaces · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • motion of a set by the curvature of its boundary · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Neck Pinching Dynamics under Mean Curvature Flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • On the dynamics of formation of generic singularities of mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • On the existence of mean curvature flow with transport term · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • On the mean convexity of a space-and-time neighborhood of generic singularities formed by mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • On the Mean Curvature Evolution of Two-Convex Hypersurfaces · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Partial regularity of mean-convex hypersurfaces flowing by mean curvature · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Physics and Geometry of Flow under Curvature: Singularity Formation, Minimal Surfaces, Geodesics, and Surface Tension · Sethian–Chopp, in Motion by Mean Curvature (1994), pp. 168–185; author scanReviewed for the Chopp catalogue addition; scope and exclusions in documentation/chopp-review.md.
  • Precise asymptotics near a pinched disk singularity formed by mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Problems in Mean Curvature Flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Quantitative Stratification and the Regularity of Mean Curvature Flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Rate of convergence of the mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Regularity of elliptic and parabolic systems · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Regularity of elliptic and parabolic systems · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Regularity of elliptic and parabolic systems · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Regularity of the level set flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Regularity Theory for Mean Curvature Flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Remarks on the self-shrinking Clifford torus · Supplied PDF; see cited edition where reviewedSelected passages used for the special-solution catalogue; property lists are limited to the cited sources.
  • Rigidity of generic singularities of mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Sharp entropy bounds for self-shrinkers in mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Sharp estimates for mean curvature flow of graphs · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Sharp lower bounds on density for area-minimizing cones · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Shortening space curves and flow through singularities · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Singularities of Mean Curvature Flow of Surfaces · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Singularities of the curve shrinking flow for space curves · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Singularity profile in the mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Smooth compactness of self-shrinkers · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Stability and area growth of lambda-hypersurfaces · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Stability of neckpinch singularities · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Starshaped hypersurfaces and the mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Stratification of minimal surfaces, mean curvature flows, and harmonic maps · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Subsequent singularities in mean-convex mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • The heat equation shrinking convex plane curves · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • The motion of a surface by its mean curvature · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • The nature of singularities in mean curvature flow of mean-convex sets · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • The round sphere minimizes entropy among closed self-shrinkers · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • The singular set of mean curvature flow with generic singularities · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • The size of the singular set in mean curvature flow of mean-convex sets · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • The topology of hypersurfaces moving by mean curvature · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • The volume preserving mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Topological change in mean convex mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Uniqueness and pseudolocality theorems of the mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Uniqueness of asymptotically conical tangent flows · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Uniqueness of blowups and Łojasiewicz inequalities · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Uniqueness of compact tangent flows in mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Uniqueness of self-similar shrinkers with asymptotically conical ends · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Uniqueness of self-similar shrinkers with asymptotically cylindrical ends · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Universality in mean curvature flow neckpinches · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.
  • Width and mean curvature flow · Supplied PDF; see cited edition where reviewedInventoried for future study; no incorporation implied.

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