Uniqueness of Tangent Cones for Area-Minimizing Integral Currents in Higher Codimension

Uniqueness of Tangent Cones for Area-Minimizing Integral Currents in Higher Codimension

Definitions and mathematical statement

An integral $m$-current represents an oriented rectifiable set with integer multiplicity and locally finite mass, with integral boundary. It is area-minimizing if no compactly supported competitor with the same boundary has smaller mass. For an interior point $p\in\operatorname{spt}T\setminus\operatorname{spt}\partial T$ and $r>0$, define \[ T_{p,r}=(\eta_{p,r})_\#T, \qquad\eta_{p,r}(x)=\frac{x-p}{r}. \] A tangent cone is a local weak limit along some $r_j\downarrow0$; weak convergence tests against compactly supported smooth differential forms. For $m\ge3$ and codimension $n\ge2$, the conjecture asserts that every such subsequential limit is the same current at every interior point. It is uniqueness of the full oriented multiplicity-bearing tangent current, not merely of its support or density, and concerns minimizing currents rather than arbitrary stationary varifolds.

Formulation and concept references: [S1].

Short English statement

At every interior point of an area-minimizing integral current in higher codimension, do all infinitesimal blow-up sequences converge to the same tangent cone?

Research attempt

Scope, attribution, and source check.

This unresolved attempt was prepared by GPT-6 Astra Ultra on 27 September 2026 using the primary account [S1], explicit blow-up calculations and checks of the quantifiers. The target is uniqueness of the oriented integral current, including multiplicity, at every interior point. The source describes uniqueness and decay results on specified strata and recalls the two-dimensional theorem. These do not establish the assertion at every point in dimensions $m\geq3$. The attempt isolates a summable-scale criterion and constructs a nonstationary current showing why density and rectifiability alone cannot replace it.

What monotonicity actually controls.

Translate the point to the origin and remain inside a ball disjoint from the boundary. The varifold associated with a minimizing current is stationary. Write $\Theta(r)=r^{-m}\|T\|(B_r)$. With this normalization the Euclidean monotonicity identity is \begin{equation}\label{wsref-0-61-34-31-38-3a-6d-6f-6e-6f} \Theta(R)-\Theta(r)= \int_{B_R\setminus B_r} \frac{|x^\perp|^2}{|x|^{m+2}}\,d\|T\|(x). \end{equation} The identity holds at admissible radii and extends by limits. Here $x^\perp$ is normal to the approximate tangent plane. Consequently $\Theta(0)$ exists and the radial tilt energy is finite near zero.

For $r_j=2^{-j}r_0$, put $e_j=\Theta(r_j)-\Theta(r_{j+1})\geq0$. Then \[ \sum_{j\geq0}e_j=\Theta(r_0)-\Theta(0)<\infty. \] This scalar energy statement does not imply that the cones selected at successive scales have summable total change. Even an estimate of angular change by $\sqrt{e_j}$, if justified in the relevant setting, would require summability of those square roots. For example $e_j=(j+1)^{-2}$ is summable and $\sum_j\sqrt{e_j}$ diverges. Rotations by $1/(j+1)$ can keep turning while their squared angular increments have finite sum. Applying Cauchy--Schwarz over infinitely many scales would introduce the square root of an infinite number of scales, not a finite bound.

A precise Cauchy criterion for full currents.

On locally mass-bounded currents choose a metric $d$ for local weak convergence using a countable dense family of compactly supported smooth test forms. A weighted sum of $\min\{1,|(S-T)(\omega_k)|\}$ suffices, with the family dense on every compact set. Uniform local mass bounds make convergence on the family equivalent to convergence against every test form.

Suppose \begin{equation}\label{wsref-0-61-34-31-38-3a-6f-73-63-69-6c-6c-61-74-69-6f-6e} \sup_{r,s\in[r_{j+1},r_j]}d(T_{0,r},T_{0,s})\leq\eta_j, \qquad \sum_j\eta_j<\infty. \end{equation} Then every blow-up sequence has the same limit. Indeed the dyadic sequence is Cauchy, since successive distances are at most $\eta_j$. Local compactness of minimizing integral currents supplies a convergent subsequence; hence the whole dyadic sequence converges to its limit $C$. For an arbitrary intermediate radius, \[ d(T_{0,r},C)\leq\eta_j+d(T_{0,r_j},C)\longrightarrow0 \quad\text{if }r\in[r_{j+1},r_j]. \] This proves convergence through all radii, not merely along a chosen discrete sequence.

An actual comparison $\eta_j\leq A\sqrt{e_j}$, together with $e_j\leq B2^{-\beta j}$ for $\beta>0$, would be a sufficient package: the resulting geometric series converges. Neither that comparison in this metric nor such power decay is established here for every singular point. They are additional hypotheses. Monotonicity alone supplies only the sum of $e_j$.

A complete regular-point calculation.

Suppose near zero the current is $Q\lbrack\!\lbrack\operatorname{graph}u\rbrack\!\rbrack$ over an oriented $m$-plane, with $u$ of class $C^1$, $u(0)=0$ and $Du(0)=0$. At scale $r$ the graphing function is $u_r(y)=u(ry)/r$. On fixed compact subsets, \[ \sup|u_r|\longrightarrow0,\qquad \sup|Du_r|=\sup|Du(ry)|\longrightarrow0. \] Parametrized integrals of compactly supported test forms converge to those on the plane, retaining its orientation and the integer coefficient $Q$. The tangent current is uniquely $Q\lbrack\!\lbrack P\rbrack\!\rbrack$. If $|Du(x)|\leq C|x|^\alpha$, the same calculation gives a power rate for fixed test-form norms and exhibits the summable-scale mechanism explicitly.

This argument assumes a genuine smooth graph near the point. A tangent plane with multiplicity $Q$ at a singular point does not automatically provide such a graph. Replacing that hypothesis by its desired conclusion would conceal the regularity gap.

A slowly rotating current as a failure test.

For $0<r<r_0<1$, let \[ \phi(r)=\log\log(e/r),\qquad \gamma(s)=s(\cos\phi(|s|),\sin\phi(|s|)) \quad(s\ne0),\qquad\gamma(0)=0. \] The parametrization is Lipschitz, since \[ |\gamma'(s)|=\sqrt{1+\log(e/|s|)^{-2}} \] is bounded. It defines an integral one-current with no boundary at zero. Because $|\gamma(s)|=|s|$, its density is \[ \frac{\|\gamma_\#\lbrack\!\lbrack(-r_0,r_0)\rbrack\!\rbrack\|(B_r)}{2r} =\frac1r\int_0^r\sqrt{1+\log(e/s)^{-2}}\,ds \longrightarrow1. \] The scalar density therefore agrees with that of a multiplicity-one line.

Choose $r_j\downarrow0$ for which $\phi(r_j)$ tends modulo $2\pi$ to $\theta$. For fixed $s\ne0$, \[ \phi(r_j|s|)-\phi(r_j) =\log\left(1+\frac{\log(1/|s|)}{\log(e/r_j)}\right) \longrightarrow0. \] The rescaled parametrizations and derivatives converge away from zero to $s\mapsto s(\cos\theta,\sin\theta)$. Their uniform derivative bound permits dominated convergence in test-form integrals across zero. A test form supported in $B_R$ only samples parameters $|s|\leq R$, so no parameter tail is lost. The tangent current is the oriented line of angle $\theta$, and different angle subsequences give different tangent currents.

This is deliberately not a minimizing example. The tangent angle changes along every punctured branch, giving nonzero curvature and a nonzero first variation for a suitable compactly supported normal variation away from zero. Thus it is not even stationary. Its product with $\mathbb R^{m-1}$, with one extra zero ambient coordinate if necessary, has the same tangent nonuniqueness in the dimensional range of the target, but still fails stationarity. It does not satisfy the full hypotheses of (a418:mono). The example tests a shortcut and is not a counterexample to the conjecture.

Density, support and exceptional points are different issues.

All tangent cones have the same density, but distinct oriented cones may share one density. Even equality of support would not alone identify orientations and multiplicities. A selection argument must compare currents or quantities controlling their test-form evaluations, as in (a418:oscillation); density is not such a metric.

Likewise uniqueness at almost every point of a rectifiable singular stratum does not settle every interior point. A full-measure subset may leave exceptional points at which the available decay theorem says nothing. The universal quantifier in the target cannot discard those points. This is why the strong stratified results described in [S1] do not by themselves close the final step here.

Verification, first gap, and outcome.

The Cauchy argument includes every intermediate radius, and the rotating example is checked as an oriented current rather than as a picture of supports. Its speed is bounded while its total turning diverges, matching the distinction between squared increments and their sum. At regular points the multiplicity is preserved under every rescaling.

The first missing general input is a sufficiently strong decay or cone-selection estimate at each singular point. A concrete continuation would seek the within-annulus control in (a418:oscillation), together with a summability mechanism stronger than the scalar monotonicity budget. UNRESOLVED. Regular graph uniqueness and the conditional scale criterion are proved. The explicit nonunique example violates the variational hypothesis, and no minimizing counterexample or general uniqueness theorem is claimed.

Sources

• S1 Skorobogatova, Singularities of area-minimizing integral currents: recent progress and open directions, Seminaire Laurent Schwartz EDP et applications $2025$. https://proceedings.centre-mersenne.org/item/10.5802/slsedp.175.pdf. Formulation source.