When Two Singularities Merge: Variational Relic Geometry

When two singularities approach each other and eventually occupy the same point, we count one location where there were previously two. This seems like a natural way to describe the collision.

For a family of energy-minimizing geometric objects, however, this simple event raises a question. The fields can converge strongly, together with their first derivatives. Their boundary values can remain smooth throughout. Yet the number of singular locations can still drop from two to one. Some geometric information survives the limit that counting locations no longer detects.

We use relic for a surviving local structure carried by an energy-minimizing map, recorded through its charge, multiplicity, and density. What survives depends on the target, the energy, and the allowed variations.

There is an example in which we can calculate the entire process. It also suggests a way to study relics: determine which local models actually minimize energy, understand how those models combine, and find a notion of counting that survives their collisions.

1. Two relics, one location

Start with a familiar direction field: place a unit arrow at every point in three-dimensional space. Rapid changes between neighboring arrows cost energy, measured by the integral of the squared gradient. Once the arrows on the boundary of a ball are fixed, we can ask for the least expensive arrangement inside.

A classical example points directly outward from the center:

This is the hedgehog. There is no way to assign a direction at the center that makes the field continuous. Approach from the right, and the arrow points right; approach from above, and it points up. Every proposed central arrow disagrees with some approach direction.

The singularity nevertheless has finite energy. At distance \(r\) from the center, the gradient has size about \(1/r\), so its square is about \(1/r^2\). The volume element of a three-dimensional spherical shell contributes a compensating factor of \(r^2\). Consequently, the energy inside a ball of radius \(r\) is proportional to \(r\). Allowing such finite-energy singularities is natural for this variational problem. For its spherical boundary values, the hedgehog does indeed have the least energy among all fillings.

Now place two arrows at each point, one red and one blue. Each field varies independently, and the total energy is the sum of their energies. There is no additional coupling term. If each arrangement minimizes its own energy, their pair minimizes the total.

Put the red hedgehog at \(te_1\) and the blue one at \(-te_1\), where \(e_1\) is a horizontal unit vector and \(0<t<1/2\). Restrict attention to the unit ball. The combined field is

Every \(U_t\) minimizes energy for its own boundary values, which vary smoothly with \(t\). There is a direct reason that a translated hedgehog remains minimizing on this ball. If a cheaper replacement existed there, we could glue it into a sufficiently large ball centered at the translated singularity. That would produce a cheaper filling for the original hedgehog and contradict its minimality.

As \(t\) tends to zero, the two centers meet and the limiting field is \(U_0=(h,h)\). The fields and their first derivatives converge in the square-integrable sense. Since the boundary stays away from the singularities, convergence there holds with control of derivatives of every order. None of this prevents two singular locations from becoming one.

At the center, however, neither arrow field admits a continuous extension. The red relic and the blue relic both survive, now at the same location.

2. Reading relic multiplicity from energy density

In three dimensions, the relic measure records both location and multiplicity. For finitely many distinct locations \(x_i\), write

Here \(\delta_{x_i}\) is a unit point mass, and \(m_i\) records the relic multiplicity at \(x_i\). In this unit-weight product, each elementary component contributes one unit. The collision becomes

Before the collision, each of two locations carries one relic. Afterward, one location carries two. Tested against any continuous function, the sum of the readings at the two moving centers tends to twice the reading at the origin.

The colored arrows make this interpretation visible. Energy provides an intrinsic way to recover the same multiplicity. At a singular point \(p\), divide the energy in a small ball by its radius and take the limit:

This is the three-dimensional energy density. Let \(\Theta_0\) be the density of one unit hedgehog. Each separated center has density \(\Theta_0\). When the centers coincide, the two component energies add, giving density \(2\Theta_0\). The normalized ratio \(\Theta_U(p)/\Theta_0\) is the relic density in this unit-weight model; it equals the relic multiplicity at that location.

The scaling matters. The hedgehog’s energy inside \(B_r(p)\) tends to zero with \(r\); there is no fixed amount of energy concentrated as a point mass at its center. Multiplicity comes from the ratio that remains as the energy shrinks. It records local information that an unweighted count of locations loses.

To understand how far this picture extends, we need to know which singular models minimize energy in the first place.

3. What a boundary filling can save

Take a smooth map \(g\) on the boundary of the unit ball. Extending its value constantly along each radius produces a radial cone \(C_g\). All the boundary’s directional variation persists down to the center.

Other fillings can rearrange the interior. Singularities may move and configurations may change. We require only the same boundary values, finite energy, and values in the prescribed target space \(Y\).

Let \(m_Y(g)\) be the least possible filling energy, and define the filling deficit by

The deficit measures how much energy can be saved by rearranging the interior while holding the boundary fixed. It is nonnegative because the radial cone is itself an admissible filling. A zero deficit means that the cone minimizes energy. A positive deficit exhibits room for improvement and excludes the cone as a minimizing model.

This gives a concrete approach to complicated singularities: construct fillings and compare their energies. Because a competitor may change the entire interior configuration, it can reveal improvements that are hard to see from the original cone alone.

For independent components, the comparison obeys an exact addition law. Suppose the target is a product of spaces \(Y_a\), with total energy the sum of the component energies weighted by fixed numbers \(w_a>0\). Then

Every product filling costs at least the sum of the component infima. Conversely, combining nearly optimal component fillings approaches that sum. A product cone therefore minimizes energy exactly when every component cone does.

4. Why only the round projective hedgehog survives

Complex projective space \(\mathbb CP^N\) parametrizes complex lines in \(\mathbb C^{N+1}\). We use its standard Fubini–Study metric, with any overall rescaling of a product factor represented by its positive weight. A projective line \(\mathbb CP^1\), after the appropriate metric scaling, is a round sphere.

Filling comparisons, combined with the structure of projective harmonic spheres, give a classification: every nonconstant three-dimensional minimizing radial cone into \(\mathbb CP^N\) is a unit hedgehog on a round projective line. Its two possible topological orientations have charge \(+1\) and \(-1\).

This answers the classification question posed by Fujioka in 1993. He proved that round projective-line hedgehogs minimize energy and ruled out holomorphic candidates of degree greater than two. Nonholomorphic candidates and degree-two holomorphic cones required further filling comparisons.

The main mechanism is a conflict between two energy bounds. Let \(e_*\) be the unit-ball energy of a unit projective-line hedgehog. Let \(\mathbb CP^d\) be the smallest projective subspace containing the image of the candidate sphere map \(g\). In three dimensions, the unit-ball energy of its radial cone equals the spherical energy of \(g\); write both as \(E(g)\). Minimality of the cone requires \(g\) to be harmonic: its first energy variation must vanish.

First consider harmonic spheres that are neither holomorphic nor antiholomorphic. Then \(d\ge2\). The classical Eells–Wood harmonic sequence expresses their energy through two neighboring integer degrees. Three-term recurrence inequalities for those degrees yield

A filling construction supplies the opposing bound. Keep a small inner core constant, and contract the surrounding shell along projective directions while preserving \(g\) on the outer boundary. An individual contraction direction can be difficult to estimate. Averaging over unitary rotations preserves the angular energy exactly and reduces the radial transition cost to an explicit constant \(K_d\).

If the constant core has radius \(\varepsilon\), the calculation gives

An upper bound on the average guarantees at least one filling whose energy is no larger. These competitors may have finite-energy singularities. At exceptional directions in the construction, checking finite energy and the boundary trace is essential to ensuring that the gluing remains admissible.

If the original cone minimized energy, then \(m_{\mathbb CP^d}(g)=E(g)\). Substituting into the filling estimate, canceling common terms, and letting \(\varepsilon\) tend to zero would give \(E(g)\le K_d\). The candidate would therefore have to satisfy

a contradiction. The substantive work lies in establishing the harmonic-sequence lower bound and the averaged-filling upper bound. Once both are available, they exclude every candidate that is neither holomorphic nor antiholomorphic.

The remaining holomorphic case has degree two. For a conic spanning a projective plane, the first-order balance condition at the center constrains its parametrization. A suitable direction then allows a surgery that splits the configuration into two nearby centers. An outer correction restores the boundary values exactly. For each such conic, the resulting competitor satisfies

The interior saving is first order; the boundary correction costs only second order. A sufficiently small split therefore lowers the energy. This comparison takes place inside a single projective target and requires its own two-center energy calculation. It does not follow from independence of product components. Degree-two covers whose image lies in a projective line are excluded by the classical sphere classification of Brezis–Coron–Lieb.

Fujioka’s theorem excludes degrees greater than two. These comparisons exclude degree two. At degree one, nonround parametrizations fail the center balance condition, while round ones survive by Fujioka’s minimality theorem. Complex conjugation gives the antiholomorphic case, and constant maps are plainly minimizing. This completes the classification; the full argument is given in Minimizing cones in complex projective spaces.

5. Elementary and composite relics

As \(N\) grows, \(\mathbb CP^N\) offers more possible states, yet every nonconstant minimizing cone still uses just one round projective line.

For a product of projective spaces, the addition law now gives all minimizing cones. Each component is either constant or a unit hedgehog. Each unit hedgehog is an elementary relic. Several independent hedgehogs sharing a center form a composite relic, whose multiplicity counts their contributions.

The relic charge is the model’s topological charge. It can be decomposable without compromising minimality. The double hedgehog has charge \((1,1)\), which decomposes as \((1,0)+(0,1)\). Independence of the component energies allows both relics to occupy the same center. Minimality is fully compatible with this composite charge.

The distinction between this product construction and the preceding conic surgery is crucial. The red and blue hedgehogs belong to independent factors. The unstable quadratic conic is a configuration within one projective factor, where a new interior filling can lower its energy.

6. From tangent cones to stable singular measures

The classification also yields stability of the singular-set measure for minimizing maps into any \(\mathbb CP^N\), under the specified strong convergence assumptions. This is a projective extension of the stability method of Mazowiecka–Miśkiewicz–Schikorra. Their work established stability for the \(S^2\) target, together with the compactness, boundary control, and covering methods used here. The projective cone classification supplies the additional target-specific information.

Precisely, let \(\Omega\subset\mathbb R^m\) be a bounded \(C^1\) domain, with \(m\ge3\). Suppose \(u_j\) and \(u\) minimize energy for their respective boundary values. Assume strong convergence in \(W^{1,2}(\Omega)\), and strong convergence of the boundary traces in \(W^{1,m-1}(\partial\Omega)\). The first condition controls the square-integrable errors of both fields and first derivatives; the second also controls their variation along the boundary.

Writing \(k=m-3\), the singular-set measures converge weakly:

with convergence of total mass as well. Here \(\mathcal H^k\) denotes \(k\)-dimensional Hausdorff measure. In a three-dimensional domain, \(k=0\), so this is singular-point counting. In higher dimensions, it measures the length, area, or corresponding higher-dimensional size of the singular set.

The proof breaks the singular set into pieces whose measure can be controlled sharply. On its highest-dimensional stratum, the blow-up models are unit hedgehogs extended constantly along another \(k\) directions. These cylindrical models share a common energy density and have transverse charge \(+1\) or \(-1\). Among cylindrical minimizing cones, they occupy the lowest nonzero density level.

These facts allow Miśkiewicz’s flatness theorem to apply: sufficient closeness to such a model at one scale persists in smaller balls centered on the singular set. The next step is to turn that geometric closeness into a quantitative statement about measure.

7. Turning geometric closeness into measure control

Under the appropriate flatness and energy conditions in a slightly larger ball, the singular measure in the smaller ball satisfies

Greater closeness makes \(\eta>0\) arbitrarily small. The quantity \(\omega_k r^k\) is the volume of a flat \(k\)-dimensional disk of radius \(r\), with the convention \(\omega_0=1\).

The lower bound comes from topology. Slice transversely to the model with three-dimensional balls. Their two-dimensional boundary spheres carry unit charge, forcing a singularity inside each slice. Orthogonal projection of the singular set onto the model’s \(k\)-plane consequently covers almost the entire disk. Projection cannot increase measure, so the singular set must have nearly the disk’s measure or more.

For the upper bound, one must control bending and extra layers at very small scales. Naber–Valtorta’s quantitative estimates bound deviations from flatness by changes in the rescaled energy across scales. Energy monotonicity, together with proximity to the lowest density, keeps the accumulated changes small. Their rectifiable Reifenberg theorem then gives the sharp upper bound. These are the analytic tools that turn geometric closeness into measure closeness; see their original paper.

In three dimensions, the estimate is particularly concrete. Choose \(\eta<1\). The singular-point count is an integer between \(1-\eta\) and \(1+\eta\), so it must equal one. A minimizing map sufficiently close to a unit hedgehog retains exactly one singularity in the small ball.

To pass from local control to the whole domain, cover most of the limiting singular set by finitely many such balls. Place the remainder inside balls of radii \(r_i\) with \(\sum_i r_i^k<\eta\). General singular-measure bounds ensure that these remainder balls contribute at most \(C\eta\) in total. In three dimensions, neighborhoods of the isolated singular points suffice, and no remainder family is needed.

Strong convergence keeps the principal-ball estimates valid for \(u_j\). Strong convergence of the boundary traces, through boundary regularity, provides a common boundary neighborhood free of singularities. Comparing the principal balls, controlling the remainder, and letting the errors vanish gives convergence of the singular measures and their total masses.

Apply this result separately to each factor of an independent projective product. Each component’s singular measure converges, so any fixed positive weighted sum also converges, including its total mass. The product measure counts component relics separately, with their prescribed weights, including when their locations coincide. The multiplicity measure of the double hedgehog is the simplest example.

This makes a sharp distinction in three dimensions. Singularities in different components may meet, adding their multiplicities at one location. Within a single component, however, the assumptions ensure that, sufficiently near the limit, singularities correspond one by one. Their locations converge, and the signs of their unit charges are preserved. Two singularities in the same component cannot merge into one or disappear together under this convergence. Conservation of the signed sum of charges alone would not imply this; minimality and the convergence assumptions control the absolute number of relics.

8. Variational Relic Geometry

We call this viewpoint Variational Relic Geometry. Boundary fillings select the local minimizing models; their classification identifies the elementary relics; energy density and singular-set measures connect their combinations and collisions to global control.

The double-hedgehog family realizes relic fusion; following the same family backward realizes relic splitting. Both preserve multiplicity here because the two component energies are independent.

The next assumption to change is independence. Enlarging the state space so that previously separate degrees of freedom can interact gives an interior filling new possibilities. A double hedgehog that minimizes energy in the product target may then acquire a cheaper competitor. The product addition law supplies a precise baseline, and the filling deficit tests what the new directions change.

Which interactions allow several relics to coexist stably, and which make a previously minimizing combination unstable? Answering that requires calculating the new filling costs. The elementary picture of two singularities becoming one provides a computable reference point for those questions.