we-solve (beta)

Math-2680 · mathematics · Geometry

Codimension-one singular bound for stationary integral varifolds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For all positive m,n and every stationary integral m-varifold V in an open subset of $R^{m+n}$, is $dim_H$ $Sing(V)\leq m-1$? Sing(V) excludes points with a neighborhood on which V is a constant positive integer multiple of a smooth embedded minimal m-submanifold.

Short English statement

For all positive m,n and every stationary integral m-varifold V in an open subset of $R^{m+n}$, is $dim_H$ $Sing(V)\leq m-1$? Sing(V) excludes points with a neighborhood on which V is a constant positive integer multiple of a smooth embedded minimal m-submanifold.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2680-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2680

Codimension-one varifold singular bound: For a stationary integral varifold of dimension $m$, the refinement claims a singular-set Hausdorff bound of $m-1$. The Euclidean ambient dimension can be any larger dimension, and the assertion begins at $m=1$. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Codimension-one singular bound for stationary integral varifolds

OpenAI's claim

Codimension-one varifold singular bound: For a stationary integral varifold of dimension $m$, the refinement claims a singular-set Hausdorff bound of $m-1$. The Euclidean ambient dimension can be any larger dimension, and the assertion begins at $m=1$.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $V$ be a stationary integral $m$-varifold in open $U\subset\mathbb R^{m+n}$, with $m,n\ge1$. Its weight has the form $\mu=\theta\mathcal H^m|_E$ with integer multiplicity, and stationarity requires $\int\operatorname{tr}(P_{T_xV}DX)\,d\mu=0$ for every compactly supported $C^1$ ambient vector field. Define singular points as support points without a neighborhood equal to one smooth embedded minimal submanifold with constant positive integer multiplicity. The claimed bound is \[\dim_H\operatorname{Sing}V\le m-1.\] Stability, minimization and a codimension-one assumption are not required.

Outline of the argument

Codimension-one varifold singular bound: Smooth reference graphs are fitted at prescribed flat scales, with controlled frequency intervals covering transitions between scales. Centered heights are normalized so that their second moments remain nonzero while their first moments vanish. The limiting height and small mean-curvature errors provide the dimension-reduction estimate needed to sharpen the earlier almost-everywhere result.

Formal verification

Codimension-one varifold singular bound: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

Go to contribution →