# Superforms, supercurrents, minimal manifolds and Riemannian geometry

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## Abstract

Supercurrents, as introduced by Lagerberg, were mainly motivated as a way to study tropical varieties. Here we will associate a supercurrent to any smooth submanifold of \({\mathbb {R}}^n\). Positive supercurrents resemble positive currents in complex analysis, but depend on a choice of scalar product on \({\mathbb {R}}^n\) and reflect the induced Riemannian structure on the submanifold. In this way we can use techniques from complex analysis to study real submanifolds. We illustrate the idea by giving area estimates of minimal manifolds and a monotonicity property of the mean curvature flow. We also use the formalism to give a relatively short proof of Weyl’s tube formula.

## 1 Introduction

A *superform* on \({\mathbb {R}}^n\) is defined as a differential form on \({\mathbb C}^n\) whose coefficients do not depend on the imaginary part of the variable. The dual of the space of superforms (with coefficients compactly supported in \({\mathbb {R}}^n\) and with the usual topology from the theory of distributions), is the space of *supercurrents*. Superforms and supercurrents were introduced by Lagerberg (2012), as a way to study tropical varieties. A tropical variety in \({\mathbb {R}}^n\) defines a *d*-closed, positive, supercurrent of integration, and conversely any such supercurrent defines a tropical variety, given a condition on the dimension of the support. (See also the work of Babaee (2014), for related work using standard currents on \(({\mathbb C}^*)^n\) instead of supercurrents.)

*M*, of \({\mathbb {R}}^n\) a supercurrent, \([M]_s\), with the aim to apply methods from complex analysis to real manifolds. These supercurrents are

*d*-closed only if the manifold is a linear subspace, but \(d[M]_s\) is given by an explicit formula involving the second fundamental form of

*M*. As a result, it turns out that

*M*is minimal if and only if

*m*is the dimension of

*M*and \(\beta \) is the Euclidean Kähler form on \({\mathbb C}^n\) (Corollary 5.2). Thus, minimality is characterized by a rather simple linear equation, which suggests a generalization of minimal manifolds to minimal ‘supercurrents’. This is of course similar to the use of (classical) currents and varifolds in the theory of minimal manifolds, but has the extra feature of a bidegree, as in complex analysis. With this we can imitate Lelong’s method for positive closed currents to prove e. g. the monotonicity formula for minimal manifolds, and a volume estimate that generalizes a recent result of Brendle and Hung (2017) (Theorem 6.3). We also obtain a result on removable singularities for minimal manifolds along the lines of the El Mir–Skoda theorem from complex analysis (Theorem 7.2), and a formula for the variation of the volume under the mean curvature flow (Theorem 8.2). ( I take the opportunity to thank Duong Phong for suggesting to apply the formalism to the mean curvature flow.)

After that we give an expression of the Riemann curvature tensor of *M* as a superform. This is basically a rewrite of Gauß’s formula. We apply it in the last section, to give a rather short proof of Weyl’s tube theorem, Weyl (1939). The proof is in essence the same as Weyl’s proof, but we have included it, hoping to show that the superformalism is useful in computations.

Finally I would like to thank an anonymous referee for a very careful reading of my manuscript, spotting many errors and suggesting several improvements.

## 2 Preliminaries

*E*be an

*n*-dimensional vector space over \({\mathbb {R}}\). Thus

*E*can be identified with \({\mathbb {R}}^n\), but at some points it will be convenient not to fix a basis. We define the ’superspace’ of

*E*to be

*E*is a differential form on \(E_s\) that is invariant under translation in the \(E_1\)-variable. If \(x=(x_1, \ldots x_n)\) and \(\xi =(\xi _1, \ldots \xi _n)\) are coordinates on \(E_0\) and \(E_1\) respectively a superform can then be written

*a*has bidegree (

*p*,

*q*) if the lengths of the multiindices in (2.1) satisfy \(|I|=p\) and \(|J|=q\). (Notice that this is not the same bigrading as in the complex case.) With these conventions, a superform of bidegree (0, 0) can be identified with a function on

*E*, since it does not depend on \(\xi \). So, a ‘superfunction’ on \(E_s\) is a function on

*E*.

We also equip \(T^*(E_s)=T^*(E_0)\oplus T^*(E_1)\) with a complex structure *J*, such that *J* maps \(T^*(E_0)\) to \(T^*(E_1)\) and vice versa. (Here our definitions differ from Lagerberg’s, who considers instead maps from \(T^*(E_0)\) to \(T^*(E_1)\) satisfying \(J^2=1\).) In the sequel we will only consider bases of \(E_0\) and \(E_1\) such that \(J(dx_i)=d\xi _i\), and therefore \(J(d\xi _i)=-dx_i\). We then extend *J* to act on forms of arbitrary bidegree so that \(J(a\wedge b)=J(a) \wedge J(b)\). When *a* is of bidegree (*p*, 0) we sometimes write \(J(a)=a^{\#}\).

*a*is

*E*and \({d^{\#}}\) is just \(d^c= i(\bar{\partial }-\partial )\), but we write \({d^{\#}}\) to emphasize that it acts only on superforms, i. e. forms not depending on \(\xi \). Note that we also have

*a*.

We also suppose given a scalar product on *E* and extend it to \(E_s\) so that it is invariant under *J* and \(E_0\perp E_1\). Thus, if \(dx_i\) are orthonormal, \(d\xi _i=dx_i^{\#}\) are orthonormal on \(E_1\) and \(dx_i, d\xi _i\) are orthonormal on \(E_s\).

*a*is a form of maximal bidegree (

*n*,

*n*), we write \(a=a_0 dx\wedge d\xi \) with \(dx=dx_1\wedge \ldots dx_n, d\xi =dx^{\#}\) and put

*define*

*a*is given by (2.2) and \(dx_i\) are orthonormal, we get

*x*and \(\xi \) both change sign, or directly from (2.3).

Let us briefly compare superintegration to classical integration over the complexification. The first problem with classical integration is of course that the classical integral over \(E_s\) would always be divergent since \(a_0\) does not depend on \(\xi \). This could be overcome by replacing \(E_1\) by its quotient by a lattice, so that we would replace \(E_s\) by \(E_0\times T^n\), where \(T^n\) is a torus. The reason that does not work here is that we will later want to integrate forms of lower bidegree over linear subspaces, and these subspaces do not in general correspond to subtori, unless the subspace satisfies a rationality condition that we cannot assume to be satisfied.

*a*is of bidegree (

*n*,

*n*). Furthermore, if

*a*is compactly supported of bidegree \((n-1,n)\),

*a*and

*b*is compactly supported. Similarly, using \({d^{\#}}=\pm JdJ\) and that the superintegral is

*J*-invariant, we also have

*d*also holds for \({d^{\#}}\). From this one verifies, for example, that if \(\rho \) is a function and

*S*is of bidegree \((n-1,n-1)\), then

*p*,

*n*), i. e. is of maximal degree in \(d\xi \). For instance, if

*D*is a smoothly bounded domain in \({\mathbb {R}}^n\) and \(a=\alpha \wedge d\xi \) is of bidegree \((n-1,n)\), then

*F*be a linear subspace of \(E=E_0\) of dimension

*m*. Then its complexification is \(F_s=F\oplus J(F)\), so

*F*defines a superspace which is a complex linear subspace of \(E_s\). Restricting a superform of bidegree (

*m*,

*m*) on \(E_s\) to \(F_s\), we thus have a definition of the integral of

*a*over \(F_s\),

*p*,

*q*), or

*bidimension*\((n-p, n-q)\), is the dual of the space of smooth, compactly supported (in

*x*!), superforms of bidegree \((n-p,n-q)\). Here we use the classical notion of duals from the theory of distributions, and we say that a supercurrent is of order zero if it is continuous for the uniform topology on superforms. Note that a supercurrent of bidegree(

*n*,

*n*) is a (classical) current of top degree on

*E*, since superforms of bidegree (0, 0) are functions on

*E*. In particular, a supercurrent of top degree and order zero is a measure on

*E*(not on \(E_s\)). As usual, given coordinates, a supercurrent can be written

*function*), i. e. as acting on forms of top degree. Then (2.5) means that if

*a*is a test form of complementary bidegree

*n*,

*n*). We define the corresponding object of bidegree (0, 0), \(*\mu \), by

*F*. Choose orthonormal coordinates so that

*m*,

*m*)-form as

*a*to \(F_s\) is just \(a_0 dx_1\wedge d\xi _1 \cdots dx_m\wedge d\xi _m\), and

*F*is

*F*as a subspace of

*E*is

*F*. In the next section we shall use this to define superintegration over general smooth submanifolds of \(E={\mathbb {R}}^n\).

An important point to notice is that whereas the standard current of integration is well defined without any extra structure, the supercurrent \([F]_s\) depends on the choice of scalar product.

*T*be a supercurrent (or superform) of bidegree (

*p*,

*p*),

*T*is

*symmetric*if \(J(T)=T\), or equivalently \(T_{I J}=T_{J I}\).

*T*of bidimension (

*m*,

*m*) is (weakly) positive if

*d*-closed.

### Proposition 2.1

The wedge product between a positive (symmetric) (1, 1)-form and a positive (symmetric) form of bidegree (*p*, *p*) is again positive.

By approximation, the same thing holds for the product of (1, 1)-forms with currents of bidegree (*p*, *p*).

*F*can be written

*a*.

*V*act in the same way by contraction. Hence, e. g. \(\vec {V}\rfloor \beta =V^{\#}\).

*G*is a (local) diffeomorphism on \(E={\mathbb {R}}^n\) and \(\alpha = \sum \alpha _{I, J} dx_I\wedge d\xi _J\) we define the pull back of \(\alpha \) under

*G*as

*x*-part of the form and the \(d\xi _j\)’s are invariant. This means that we extend

*G*to a (local) diffeomorphism on \(E_s\) by leaving \(E_1\) fixed, and then take the usual pullback. Then

*n*,

*n*) and

*G*is orientation preserving.

*p*, 0) then

## 3 Relation to convex functions and tropical varieties

*I*is a finite set. We will call such functions, i. e. the maxima of a finite collection of affine functions, quasitropical polynomials, reserving the term tropical polynomials for such functions where all components of the (co)vectors \(a^i\) and the numbers \(b^i\) are integers, see Mikhalkin (2006). We may assume that all the \(a^i\) are different. Indeed, if \(a^i=a^k\) and say \(b^i> b^k\), then \(l_i>l_k\) everywhere, and we get the same function if we omit \(l_k\).

In this way \(d{d^{\#}}\phi \) describes the *tropical variety* defined by the faces \(F_l\), endowed with the multiplicity vectors \(v_l\). (Perhaps it would be more proper to talk of quasitropical variety since the multiplicity vectors are not necessarily integral.) The fact that \(d{d^{\#}}\phi \) is closed is equivalent to the *balancing condition* in tropical geometry: At a point where several faces intersect, the sum of their multiplicity vectors vanish (see Lagerberg 2012). Conversely, Lagerberg shows that a positive closed supercurrent of bidegree (1, 1) with support of dimension \(n-1\) (see Lagerberg 2012 for precise, and also more general, statements) equals \(d{d^{\#}}\phi \) for some quasitropical polynomial \(\phi \).

## 4 Supercurrents associated to general submanifolds of \({\mathbb {R}}^n\)

*M*be a smooth submanifold of \({\mathbb {R}}^n\) of dimension

*m*. Given an orientation of

*M*we get the current of integration of

*M*. Let us first assume that

*M*is a hypersurface, locally defined by an equation \(\rho =0\), where \(\rho \) is smooth and has nonvanishing gradient on

*M*. Dividing by \(|d\rho |\), we may assume that \(|d\rho |=1\) on

*M*, and we let \(n=d\rho \); it is a unit normal form on

*M*. Now it is a familiar fact that the current of integration on

*M*can be written

*M*and the Hodge star indicates that we think of it as a current of degree zero. The choice of sign of

*n*determines the orientation of

*M*. From this formula we see in particular that if \([M]\wedge v^{\#}\) is symmetric, then

*v*must be a multiple of

*n*, hence normal to

*M*(we used this at the end of the last section, with \(M=F_j\)). Moreover, if \([M]\wedge v^{\#}\) is positive,

*v*is a positive multiple of

*n*.

*M*has codimension

*p*, it is locally defined by

*p*equations \(\rho _j=0\), such that \(d\rho _j\) are linearly independent on

*M*. Replacing \(\rho _j\) by \(\sum a_{j k}\rho _k=: \rho _j'\), for a suitable matrix of functions \(a_{j k}\), we may assume that \(n_j:=d\rho _j\) are orthonormal on

*M*. Then the currents of integration on

*M*can be written

*M*is defined as

*M*by an orthogonal transformation. If \(\alpha \) is a superform on the ambient space,

*M*. Accordingly, we say that \(\alpha \) vanishes on \(M_s\) if (and only if) \(\alpha \wedge [M]_s=0\)

*M*is linear. When computing \(d[M]_s\) we will have use for the (1, 1)-forms

*F*to

*M*is the second fundamental form of

*M*. In higher codimension the \(F_j\) restricted to

*M*are the components of the vector valued second fundamental form \(\sum F_j\otimes \vec {n}_j\) with values in the normal bundle of

*M*. (Recall that \(n_j\) form an orthonormal system for the space of normal (1, 0) forms at each point.)

*M*] is

*d*-closed. This expresses \(d[M]_s\) in terms of [

*M*] but we want to write it in terms of \([M]_s\). Therefore we introduce the operator

*F*. Notice that \({\mathcal F}\) is an antiderivation. Then

### Lemma 4.1

\( n_j\rfloor F_j=0 =n_j^{\#}\rfloor F_j\) on \(M_s\), i. e. when wedged with \([M]_s\).

### Proof

*n*, i. e. the partial derivatives of \(\rho \) with respect to \(x_j\). Then

*n*is constant on

*M*, this vanishes on \(M_s\). \(\square \)

## 5 Minimal submanifolds

We start with the following computational proposition.

### Proposition 5.1

*M*is a smooth

*m*-dimensional submanifold of \({\mathbb {R}}^n\)

### Proof

*F*. This implies that

*F*’s restriction to \(M_s\), since wedging with \([M]_s\) kills all the components of

*F*and \(\beta \) in the normal directions. The traces \(tr'(F_j)=:H_j\) are the coefficients of the

*mean curvature vector*

*M*is a

*minimal manifold*, so we have proved

### Corollary 5.2

\([M]_s\wedge \beta ^{m-1}\) is a closed current if and only if *M* is a minimal submanifold.

We are therefore led to the following

### Definition

*T*of bidimension (

*m*,

*m*) is minimal if

*m*,

*m*), if it happens that there is a vector field \(\vec {V}=\sum V_j \partial /\partial x_j\) such that

*T*. As we have seen, this is the case when \(T=[M]_s\) is associated to a smooth manifold, but it certainly also holds when

*T*is strictly positive and smooth, so that \(T\wedge \beta ^m/m!\) is a strictly positive volume form. In both these cases, \(\vec {V}\) is uniquely determined, so we may speak of

*the*mean curvature vector. In Sect. 8 we shall see that when \(T=[M]_s\) is associated to a smooth manifold,

*m*,

*m*):

*u*is sufficiently smooth on \({\mathbb {R}}^n\). When \(T=[M]_s\) is the supercurrent associated to an

*m*-dimensional manifold, this is precisely the standard Dirichlet form

*u*restricted to

*M*, and \(|d_Mu|\) is the norm induced by the Euclidean metric on \({\mathbb {R}}^n\). Writing \(S_T=T\wedge \beta ^{m-1}/(m-1)!\) we have in general

*T*by

*T*is the supercurrent of a manifold, or more generally has a mean curvature vector, we can make this more explicit: First we compute

*T*has a mean curvature vector \(\mathbf {H}=\sum H_j\partial /\partial x_j\), and we let \(H^{\#}=\sum H_j d\xi _j\) be the corresponding (0, 1)- form, we have with \(\sigma =T\wedge \beta ^m/m!\),

*T*is minimal, so that \(S_T\) is closed and \(\mathbf {H}\) vanishes, we have

*T*not necessarily minimal, we get by applying (5.9) to a coordinate function, \(u=x_j\), that the components of \(\mathbf {H}\), \(H_j=\mathbf {H}(x_j)\) are given by

## 6 Volume estimates for minimal submanifolds

To prove volume estimates for minimal manifolds (or supercurrents), we will now follow the method of Lelong to prove such estimates in the complex setting. This requires one little twist since the minimal supercurrent *T* (e.g. \(T=[M]_s\)) is not closed itself; it is only \(T\wedge \beta ^{m-1}\) that is closed. This is taken care of by the following lemma.

### Lemma 6.1

*p*is an integer,

### Proof

*T*is positive of bidegree (

*m*,

*m*). This follows from Proposition (2.1) since \(|x|_\delta \) is convex. We also remark that it follows from the proof of the proposition that

*T*be a minimal current of bidimension (

*m*,

*m*), defined in a neighbourhood of the origin. Its mass in a ball of radius

*r*centered at the origin is

*T*is smooth. Then, writing \(S=T\wedge \beta ^{m-1}/m!\)

*monotonicity theorem*for minimal manifolds (see Colding and Minicozzi 1999).

### Theorem 6.2

*T*be a minimal (super)current of bidimension (

*m*,

*m*) defined in a neighbourhood of the origin. Then

*T*is the supercurrent of a minimal manifold, this says that the area of the manifold inside a ball of radius

*r*, divided by \(r^{-m}\) is nondecreasing. In analogy with the case of minimal manifolds we call

*T*at the origin. When

*T*is the supercurrent of a (smooth) minimal manifold it equals \(\omega _m\), the volume of

*m*-dimensional unit ball. (The corresponding limit for closed positive (

*m*,

*m*)-currents in complex analysis is the

*Lelong number*of the current.)

*T*smooth, but by approximation it holds for general minimal supercurrents of bidimension (

*m*,

*m*). It means in particular that the integral in the right hand side is bounded as \(\delta \rightarrow 0\), so there is a subsequence of \(\delta \)’s such that

*x*| is smooth, this measure must be

*r*is

*T*(the trace of \(d{d^{\#}}E_{m-2, 0}\)) equals a point mass at the origin of size \(\gamma _T\), plus a nonnegative contribution outside the origin. The contribution outside the origin vanishes when

*T*is the supercurrent of an

*m*-dimensional plane through the origin. This reflects the fact that \(E_{m-2,0}\) is a fundamental solution of the Laplacian then; the crucial observation is that \(E_{m-2,0}\) is always ’subharmonic on

*T*’.

*D*. Then |

*x*| is not constant on the boundary of

*D*, so instead we write \(|x|^m =w(x)\) on the boundary , where

*w*is a positive smooth function on the closure of

*D*to be chosen later. Let

*T*be a minimal supercurrent (which we tacitly take as smooth at first) in a neighbourhood of \(\bar{D}\). Then the mass of

*T*in

*D*is

*x*| by \(|x|_\delta \) and we write \(E_{m-2}\) for \(E_{m-2,0}\).) By Stokes’ theorem this equals

*S*is closed. By what we have just seen, \(I\ge w(0)\gamma _T(0)\). To see when

*II*is positive we compute (for \(m>2\))

*S*is symmetric we find that

*D*and

*w*is convex.

### Theorem 6.3

*D*be a smoothly bounded domain in \({\mathbb {R}}^n\). Let

*a*be any point in

*D*and let

*w*be a convex function on \(\bar{D}\) such that \(w=|x-a|^m\) on the boundary of

*D*. If

*T*is a minimal supercurrent in

*D*of bidimension (

*m*,

*m*), then its mass satisfies

*T*at

*a*).

(We shall see in the next section that the convexity assumption on *w* can be relaxed considerably.)

### Corollary 6.4

*M*be a minimal manifold of dimension

*m*in the unit ball which contains the point

*a*. Then the volume of

*M*satisfies

*m*-dimensional unit ball.

### Proof

*v*is convex (in fact linear), \(w:=v^{m/2}\) is also convex and the corollary follows directly from the theorem, since \(\gamma _T(a)=\omega _m\) when

*T*is the supercurrent of a (smooth) manifold. \(\square \)

## 7 Removable singularities

The techniques of the previous section can also be used to give a variant of the El Mir–Skoda theorem on extension of positive closed currents (El Mir 1984; Skoda 1982) in the setting of minimal manifolds or minimal currents. It should be stressed that here we are not aiming to prove that a minimal surface with possible singularities on a small set is actually smooth, but rather that the minimal current defined by the surface outside the singularities extends as a minimal current. This is a much simpler problem than regularity, but the results are still useful as they will allow us to apply the calculus of supercurrents to manifolds with singularities on a set of sufficiently small dimension. We refer to Harvey and Lawson (1975) for a discussion of and very precise results on removable singularities in the classical setting.

*u*to be

*m-subharmonic*if

*m*-subharmonic functions with smooth ones. If

*u*is smooth and we fix a point we can make an orthogonal change of coordinates so that

*u*is

*m*-subharmonic if and only if the sum of any

*m*-tuple of eigenvalues of \(d{d^{\#}}u\) is nonnegative at any point. We also see from (7.1) that if

*u*is smooth and

*m*-subharmonic, then

*T*is a positive supercurrent of bidegree \((n-m,n-m)\).

*m*-subharmonic, and taking limits when \(\delta \rightarrow 0\) the same thing holds for \(E_{m-2}\). Therefore, any potential

*m*-subharmonic, if \(\mu \) is a positive measure.

Kernels like \(E_{m-2}\) on \({\mathbb {R}}^n\) are called Riesz kernels, and have a well developed potential theory, following the lines of the more classical potential theory for the Newtonian kernel \(E_{n-2}\), see Landkof (1972). Thus, we have a notion of capacity \(C_{m-2}\) associated to \(E_{m-2}\) and any set of sigma-finite \((m-2)\)-dimensional Hausdorff measure has capacity zero.

We also say that a set *F* is *m*-polar is there is an *m*-subharmonic function which is equal to \(-\infty \) on *F* (and maybe elsewhere as well). By Landkof (1972), any compact set *K* with \(C_{m-2}(K)=0\) is *m*-polar, and moreover there is a measure \(\mu \) supported on *K* whose potential equals \(-\infty \) precisely on *K*. Therefore we get

### Proposition 7.1

Any compact set *K* of \(\sigma \)-finite \((m-2)\)-dimensional Hausdorff measure is *m*-polar and there is a potential *u* of a measure supported on *K* which equals \(-\infty \) on *K*.

To illustrate this, notice that it is immediate that a discrete set of points is 2-polar, and that a submanifold of dimension \((m-2)\) is *m*-polar in general. Indeed, it suffices to take \(\mu \) equal to surface measure.

### Theorem 7.2

*T*be a minimal supercurrent of bidimension (

*m*,

*m*), defined in \(B\setminus K\), where

*B*is a ball and

*K*and is compact in \({\mathbb {R}}^n\) with sigma-finite \((m-2)\)-dimensional Hausdorff measure. Assume that

*T*has finite mass

*T*, \(\tilde{T}:=\chi _{B\setminus K} T\) , is a minimal supercurrent in

*B*.

### Proof

*u*be a potential as in (7.2) with \(\mu \) supported on

*K*, which equals \(-\infty \) on

*K*. Thus,

*u*is smooth outside

*K*and

*m*-subharmonic. Let \(\chi (t)\) be an increasing smooth convex function, defined when \(t\le 0\), with \(\chi (0)=1\) and \(\chi (t)=0\) for \(t\le -1\). Put

*m*-subharmonic, \(0\le u_k < 1\), and \(u_k\) tend to 1 uniformly on compacts outside

*K*. Moreover, all \(u_k\) vanish in a neighbourhood of

*K*.

*B*. Then, by integration by parts,

*C*is a fixed constant independent of

*k*. Here we have used that \(T\wedge \beta ^{m-1}\) is closed outside of

*K*, and that \(u_k\) vanishes near

*K*. This implies that

*m*-subharmonic.

*p*(

*x*) be a smooth function on \({\mathbb {R}}\), such that \(p(x)=1\) if \(x>1/2\) and \(p(x)=0\) if \(x<1/3\), and put \(\chi _k=p(u_k)\). Then \(\chi _k\) tends to \(\chi _{B\setminus K}\) and we get if \(\psi \) is a smooth (1, 0)-form supported in

*B*,

*B*. By Cauchy’s inequality we have, choosing \(\alpha =du_k\) and \(\tau = p'(u_k)\psi \) that

*B*or \(B\setminus K\) in the left hand side since \(du_k\) and \(p'(u_k)\) vanish near

*K*.) By (7.3), the first integral on the right hand side is bounded independently of

*k*. The second integral in the right hand tends to zero by dominated convergence since \(p'(u_k)\) tends to zero on \(B\setminus K\). Hence

*d*-closed, so we are done. \(\square \)

*K*. It is easy to see, by the monotonicity theorem, that

*T*is associated with a minimal manifold of dimension

*m*that has singularities at the set

*E*the density is \(\omega _m\) (the volume of an

*m*-dimensional unit ball) at all the regular points. Therefore it is at least \(\omega _m\) at the singular points.

(A natural question in this context seems to be if minimal manifolds can be characterized within the class of minimal supercurrents by conditions on the density. For instance, if *T* is minimal and the density is constant on the support of *T*, is *T* then \(c[M]_s\) for a smooth minimal manifold *M* ?)

*w*be convex in Theorem 6.3 can be relaxed to

*m*-subharmonic, since we just need that

## 8 Variation of volume and the mean curvature flow

*M*is a smooth manifold of dimension

*m*, then

*S*is

*J*-invariant and of even degree it follows after a small computation that

*M*moves under

*the mean curvature flow*, see Colding et al. (2015), Smoczyk (1999). As in these references, we say that a family \(M_t\) of smooth submanifolds of \({\mathbb {R}}^n\) moves by the mean curvature flow if there is a vector field \(\vec {H}\) on the ambient space such that \(F_t(M_0)=M_t\), where \(F_t\) is the flow of \(-\vec {H}\), which restricts to the mean curvature vector field of \(M_t\) on each \(M_t\).

*M*] is closed.

We first note that the next proposition follows directly from (8.3) and Cartan’s formula.

### Proposition 8.1

We now let \(\sigma _t=[M_t]_s\wedge \beta _m\); the volume form of \(M_t\). (Note that this is not the same thing as \(F_{-t}^*(\sigma )\) which has total mass independent of *t*, whereas the mass of \(M_t\) changes.) The next theorem is the main result of this section.

### Theorem 8.2

*m*-dimensional submanifolds of \({\mathbb {R}}^n\), moving under the mean curvature flow, with \(M_0=M\). Let

*t*at \(t=0\) and use the proposition in the left hand side. The result is

*S*contains a factor \(n^{\#}=n_1^{\#}\wedge \ldots n_p^{\#}\). Moreover, by (8.2),

Integrating (8.5) against a smooth function \(\rho \), and using (2.4), we obtain

### Corollary 8.3

*m*, moving under the mean curvature flow. Then, if \(\rho \) is a sufficiently smooth function

*m*-subharmonic), then

To understand the intuitive meaning of the corollary, let first \(\rho \) be constant equal to 1. Then we get the well known fact that the volume of *M* decreases under the mean curvature flow. When \(\rho \) is nonconstant, convex and nonnegative we get an extra negative contribution to the derivative of the integral of \(\rho \) over \(M_t\), so, roughly speaking, \(M_t\) tends to move towards where \(\rho \) is small. The next consequence is a more precise formulation of this.

### Corollary 8.4

Under the same hypotheses as in Theorem 8.2, assume also that \(M=M_0\) lies in a convex open set \(\Omega \). Then \(M_t\) lies in \(\Omega \) for \(t>0\) as long as the mean curvature flow exists.

This follows since we may choose \(\rho \) to be zero in the convex set and strictly positive outside. Gerhard Huisken has kindly informed me that this is a well known property that follows from the parabolic maximum principle, using the formula for the mean curvature field in formula (5.11).

An open set is convex if it has a convex defining function, i. e. if there is a nonnegative convex function \(\rho \), such that \(\Omega =\{\rho <1\}\). Analogously we could define *m*-convex domains as domains that possess an *m*-subharmonic defining function. (Recall that we have defined \(\rho \) to be *m*-subharmonic if \(d{d^{\#}}\rho \wedge \beta _{m-1}\ge 0\). ) Then the last corollary also holds if \(\Omega \) is only *m*-convex, where *m* is the dimension of *M*. This condition becomes weaker when the dimension is large: For curves we need convexity, but for hypersurfaces \((n-1)\)-convexity is enough.

Formula (8.5) reflects the fact that the mean curvature flow is described by a (non linear) parabolic equation. It says that the volume forms then flow by a (linear) parabolic equation. It seems to be a natural question if there is a corresponding equation for the supercurrents \([M_t]_s\) themselves, i. e. without taking traces as we have done here. Since the minimal surface equation becomes linear in the supercurrents formalism, this is perhaps not completely unrealistic.

## 9 General submanifolds and their Riemannian geometry

In this section we shall describe the Levi-Civita connection and Riemannian curvature of a submanifold *M* of \({\mathbb {R}}^n\) in terms of the ’superstructure’.

*M*be a smooth submanifold of \({\mathbb {R}}^n\) and let \([M]_s\) be its associated supercurrent. With the notation from Sect. 3 we have

*D*and \(D^{\#}\) are antiderivations, so they satisfy the same rules of computation as

*d*and \({d^{\#}}\). If

*a*is a superform on the ambient space we get

*a*vanishes on \([M]_s\), i.e. \(a\wedge [M]_s=0\), then

*Da*and \(D^{\#} a\) also vanish on \([M]_s\), which means that

*D*and \(D^{\#}\) are well defined as operators on forms on \([M]_s\).

*p*, 0). Hence \(D=d\) on such forms and we have just retrieved the fact that the exterior derivative is well defined on submanifolds. On the other hand, \({\mathcal F}^{\#}\) involves contraction with \(n_j\) and does not vanish on forms of bidegree (

*p*, 0), so \({d^{\#}}\) and \(D^{\#}\) are different on (

*p*, 0) forms. In fact, \(D^{\#} a\) can be thought of as the Levi-Civita connection acting on

*a*. Indeed, \(D^{\#} a\) is a form of bidegree (

*p*, 1), so it can be seen as a 1-form in \(\xi \) with values in the space of (

*p*, 0)-forms in

*x*. If \(V=\sum V_j \partial /\partial x_j\) is a tangent vector to

*M*, the derivative of

*a*in the direction

*V*is

To compute its curvature we need to apply \(D^{\#}\) twice. This is done in the following proposition which is just a way of writing Gauß’s formula for the curvature in supernotation (see also Weyl (1939)).

### Proposition 9.1

*R*on

*M*is given by the symmetric (2, 2) form

*a*is of bidegree (1, 0)

If *a* is a form of bidegree (*p*, *q*) on \(M_s\) we can always extend it to a (*p*, *q*)-form on the ambient space. It is convenient to choose a special extension. We say that the extension is canonical if \(n_j\rfloor a =n_j^{\#}\rfloor a=0\) for \(j=1,\ldots p\). It is obvious that canonical extensions exist locally since \(n_j=d\rho _j\) are linearly independent and can be completed to a basis for the (1, 0) forms. We then just take any extension, write it in terms of the new basis, and throw away the terms that contain some \(n_j\) or \(n_j^{\#}\). The resulting form coincides with *a* on \(M_s\) in the sense that its wedge product with \([M]_s\) equals \(a\wedge [M]_s\).

*a*is a canonical extension of a form on \(M_s\), then \({\mathcal F}a={\mathcal F}^{\#} a=0\), so

*a*is (

*p*,

*q*) form, then

*a*has bidegree (1, 0)

### Lemma 9.2

*a*is any superform (canonical or not).

### Proof

*f*is a function. Therefore we may assume that \(a=dx_I\wedge d\xi _J\). Then \({d^{\#}}a=0\) and

*a*of any bidegree. Combined with the remark immediately before the lemma, this gives that

*a*is (1, 0), which concludes the proof of the proposition.

*M*are intrinsic, i. e. they do not depend on the embedding of

*M*in \({\mathbb {R}}^n\), since the Riemann curvature (and the metric) are intrinsic. (This is of course not true for similar integrals containing arbitrary combinations of \(F_j\).) In the next section we shall give an illustration of this.

## 10 Weyl’s tube formula

We will now use the formalism of the previous section to give a quick proof of Weyl’s tube formula (Weyl 1922; Gray 1990). The proof is not really different from the original proof, but the formalism helps to organise the formulas.

*M*be a compact submanifold of \({\mathbb {R}}^n\). We will consider \(T_r(M)\), the tube around

*M*of width

*r*, which when

*M*is without boundary is defined as

*r*is sufficiently small, \(T_r(M)\) is by the tubular neighbourhood theorem diffeomorphic to a neighbourhood of the zero section of the normal bundle

*N*(

*M*) of

*M*,

*v*in the normal bundle is a vector, normal to \(T_p(M)\) where \(p=\pi (v)\), \(\pi \) being the projection from

*N*(

*M*) to

*M*, and the diffeomorphism is

*M*is a compact manifold

*with*boundary, we

*define*\(T_r(M)\) to be the image of \(\Delta (M,r)\) under this map. Weyl’s tube formula expresses the volume of these tubes, \(|T_r(M)|\), in terms of curvature integrals over the manifold.

### Theorem 10.1

*M*be a compact

*m*-dimensional submanifold of \({\mathbb {R}}^n\) and let \(T_r(M)\) be the tube of width

*r*around

*M*. Then, for

*r*small

*M*.

Thus the theorem says first that the volume of the tubes of width *r* is a polynomial in *r* for *r* small, and moreover and most remarkably that the coefficients of the polynomial are intrinsic. Thus, if we have two isometric embeddings of the Riemannian manifold *M* into \({\mathbb {R}}^n\), the tube volumes are the same.

*M*. Then the normal bundle is trivial and isometric to \(M\times B_r(0)\), where \(B_r(0)\) is the ball in \({\mathbb {R}}^n\) of radius

*r*and center 0, via the orthonormal frame \(\vec {n_j}\). Hence the diffeomorphism

*G*described above can be written

*G*as in Sect. 2 we get

*y*,

*t*) and the Berezin integral with respect to \(\xi \). We expand the integrand by the trinomial theorem (since all three forms within the parenthesis are of degree 2 they commute), and get

*c*. What remains is therefore to compute the form valued integral

*q*in

*a*, and homogenous of order \(2q+p\) in

*r*. Hence, up to a constant, the integral in (10.2) equals

*a*of degree \(m-l\). The fact that the resulting homogenous polynomial in

*a*is given by (10.3) is therefore an algebraic identity. This must also hold when \(a_j\) are not real numbers but lie in any commutative algebra over the reals, like in our case when \(a_j=F_j\) are two-forms. Hence

## Notes

### Acknowledgements

Open access funding provided by Chalmers University of Technology.

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