Superforms, supercurrents, minimal manifolds and Riemannian geometry

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 $\R^n$. Positive supercurrents resemble positive currents in complex analysis, but depend on a choice of scalar product on $\R^n$ 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 relatively short proof of Weyl's tube formula.


INTRODUCTION
A superform on R n is defined as a differential form on 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 R n and with the usual topology from the theory of distributions), is the space of supercurrents. Superforms and supercurrents were introduced by Lagerberg , [13], as a way to study tropical varities. A tropical variety in 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, [2], for related work using standard currents on (C * ) n instead of supercurrents.) Here we extend the 'superformalism' in a different direction by associating to any smooth (or piecewise smooth) submanifold, M, of 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] s ∧ β m−1 /(m − 1)! is closed, where m is the dimension of M and β is the Euclidean Kähler form on 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, [5](Theorem 6.3). We also obtain a result on removable singularities for minimal manifolds along the lines of the 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 8.1). ( 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, [19]. 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.

PRELIMINARIES.
We start by recalling the definitions and basic properties of superforms and supercurrents, mainly following Lagerberg, [13] (and [7]), but with some modifications. Let E be an ndimensional vector space over R. Thus E can be identified with R n , but at some points it will be convenient not to fix a basis. We define the 'superspace' of E to be where we use the subscripts to indicate the first or second summand. A superform on E is a differential form on E s that is invariant under translation in the E 1 -variable. If x = (x 1 , ...x n ) and ξ = (ξ 1 , ...ξ n ) are coordinates on E 0 and E 1 respectively a superform can then be written where the coefficients do not depend on ξ. We say that a superform a has bidegree (p, q) if the length of the multiindices in (2.1) satisfy |I| = p and |J| = q. With these conventions, a superform of bidegree (0, 0) can be identified with a function on E, since it does not depend on ξ. So, a 'superfunction' on E s is a function on E.
We also equip T * (E s ) = T * (E 0 ) ⊕ 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 ) = dξ i , and therefore J(dξ i ) = −dx i .
The ordinary exterior derivative of a is and we also define E s is thus just the complexification of E and d # is just d c = i(∂ − ∂), but we write d # to emphasize that it acts only on superforms, i. e. forms not depending on ξ. When a is of bidegree (p, 0) we sometimes write J(a) = 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 ⊥ E 1 . Thus, if dx i are orthonormal, dξ i = dx # i are orthonormal on E 1 and dx i , dξ i are orthonormal on E s .
The main point in the construction of E s is the definition of integrals. If a is a form of maximal bidegree (n, n), we write a = a 0 dx ∧ dξ with dx = dx 1 ∧ ...dx n , dξ = dx # and put The integral over E 0 here is well defined as soon as we have chosen an orientation of E 0 , if we assume that a 0 has enough decrease at infinity to make the integral convergent. For the integral with respect to ξ we define if dx i are orthonormal and oriented, where c n = (−1) n(n−1)/2 . Except for the constant c n , this is the Berezin integral, [6]. The reason for introducing the factor c n is that we want the integral of to be positive if a 0 is positive in accordance with the complex case. If a is given by (2.2) and dx i are orthonormal, we get where dλ is Lebesgue measure. Note that if we change orientation, the 'superintegral' remains the same. This follows since the integrals with respect to x and ξ both change sign, or directly from (2.3). Let us briefly compare this 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 ξ. This could be overcome by replacing E 1 by its quotient by a lattice, so that we would replace E s by E 0 × T n , where T n is a torus. The reason that does not work here is that we will next 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.
Let now F be a linear subspace of E = E 0 of dimension m. Then its complexification is F s = F ⊕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 , Fs a as before.
Having defined superforms and their integrals, we now turn to supercurrents. The space of supercurrents of bidegree (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 where T I,J are distributions on E 0 . Let us explain this notation a bit more. We think of a distribution as having bidegree (0, 0) (a generalized function), i. e. as acting on forms of top degree. Then (2.4) means that if a is a test form of complementary bidegree In particular, if T I,J are locally integrable functions, then In practice, for us, the coefficients will at worst be measures. If µ is a measure, it acts on functions, and so should be regarded as a current of bidegree (n, n). We define the corresponding object of bidegree (0, 0), * µ, by * µ.α 0 dx 1 ∧ dξ 1 ∧ ...dx n ∧ dξ n = µ.α 0 .
As an example of this we consider superintegration over a linear subspace F . Choose orthonormal coordinates so that ..x n = 0} and write an (m, m)-form as Then the restriction of a to F s is just a 0 dx 1 ∧ dξ 1 ...dx m ∧ dξ m , and Fs a = F a 0 * dλ F as before. Unwinding definitions, this means that the supercurrent defined by superintegration over F is Since the standard current of integration on F as a subspace of E is ..n is a frame for the conormal bundle of F . In the next section we shall use this to define superintegration over general smooth submanifolds of E = 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.
Just as in the complex case we now say that a supercurrent T of bidimension (m, m) is (weakly) positive if T.α 1 ∧ α # 1 ∧ ...α m ∧ α # m ≥ 0 for any choice of compactly supported (1, 0)-forms α j . It is then easily verified that the supercurrent of a linear subspace is positive, and it is also immediately clear that it is d-closed.
Similarily, we say that a superform α of bidegree (n − m, n − m) is (weakly) positive if at every point. This is clearly equivalent to saying that the expression in (2.5) is positive at any point when α j are constant. When α = α jk dx j ∧ dξ k is of bidegree (1, 1), this means that the matrix of coefficients (α jk ) is positive semidefinite. At any point, such a form can be written By approximation, the same thing holds for the product of (1, 1)-forms with currents of bidegree (p, p).
We next introduce the analog of the Kähler form in R n . This is by definition Then, the volume form on F can be written in good analogy with the complex case.
In the coming sections we will frequently use contraction with a 1-form in the computations. This is well defined, since we have induced scalar products on the space of all forms, and we define contraction as the dual of exterior multiplication. If a = a j dx j is a (1, 0)-form, then a⌋β = a # = J(a), whereas a # ⌋β = −a = J(a # ). Therefore, for any 1-form a.
Similarily, contraction with a vector field is well defined. Lowering indices, V corresponds to the (1, 0)-form V = V j dx j , and V and V act in the same way by contraction. Hence, e. g. V ⌋β = V # . Finally, we will discuss how superforms transform under diffeomorphisms: If G is a (local) diffeomorphism on E = R n and α = α I,J dx I ∧ dξ J we define the pull back of α under G as i. e. the pull back is the standard pull back on the x-part of the form and the dξ 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 if α is of bidegree (n, n) and G is orientation preserving. Let now V be a vector field on R n and let G t be its flow, or one parameter family of diffeomorphisms. The classical formula of Cartan for the Lie derivative, [18], says that if α is of bidegree (p, 0) then where V ⌋ means contraction with the field V . This formula holds also for forms of general bidegree. Indeed, this follows immediately from Cartan's formula on the superspace E s , if we extend V to a vector field on E s that has no component in the second factor.

RELATION TO CONVEX FUNCTIONS AND TROPICAL VARIETIES
It is clear from the last part of the previous section that if φ is a smooth function on R n , then φ is convex if and only if dd # φ is a positive superform. By approximation, it follows that a general, possibly not smooth, function φ is convex if and only if dd # φ is a positive supercurrent.
If α is moreover positive, φ must be convex, so the positive symmetric (1, 1)-currents are precisely the ones that can be written dd # φ for some convex φ.
If φ is a smooth convex function, we can define (dd # φ) n /n! which equals The corresponding measure on R n is the Monge-Ampére measure of φ By the Bedford-Taylor theory ( [3]) this definition makes sense for general convex functions. Indeed, Bedford and Taylor define (dd c φ) n /n! for general locally bounded plurisubharmonic functions, hence in particular for convex functions on R n , considered as functions on C n that do not depend on the imaginary part of the variable. This means that for a general convex function on R n , where the Monge-Ampére measure in the right hand side is taken in the sense of Alexandrov. This follows, since the two sides coincide for smooth functions and are continuous under uniform convergence.
(To be quite honest, Bedford and Taylor [3] define (dd c φ) n as a measure on C n and prove continuity under decreasing sequences in [4], hence also for unform convergence when the functions are continuous. Since , if χ is a function of ξ with integral 1 and π is the projection from C n to R n , existence and continuity of (dd # φ) n follows.) Let us now consider convex functions of the form where 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 [15]. 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 .
Then E j is defined by a finite set of linear inequalities. Assume that one of these sets, E k has empty interior. Then we can define and get a new quasitropical polynomial which equals φ on a dense set. Hence, by continuity, φ k = φ everywhere, so we may as well omit l k in the definition of φ, and can assume that all E j have non empty interior. We therefore get a decomposition of R n as a finite union of non degenerate but possibly unbounded polyhedra. We then also have that all the E j are different, since if a j · x + b j = a k · x + b k on an open set, a j = a k which we have assumed is not the case.
It is clear that where χ E j is the characteristic function of the polyhedron E j . Since dχ E j = [∂E j ] for some choice of orientation, we get where F l is an enumeration of the faces of the polyhedron E j . These v l must be normal to F l , because dd # φ is symmetric (see the next section for this). Positivity of dd # φ implies that they point in the same direction as the normal to F l determining the orientation. In this way dd # φ 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 dd # φ 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 [13]). Conversely, Lagerberg shows that a positive closed supercurrent of bidegree (1, 1) with support of dimension n − 1 (see [13] for precise, and also more general, statements) equals dd # φ for some quasitropical polynomial φ.

SUPERCURRENTS ASSOCIATED TO GENERAL SUMANIFOLDS OF R n
Let M be a smooth submanifold of 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 ρ = 0, where ρ is smooth and has nonvanishing gradient on M. Dividing by |dρ|, we may assume that |dρ| = 1 on M, and we let n = dρ; it is a unit normal form on M. Now it is a familiar fact that the current of integration on M can be written It is clearly positive and symmetric. More generally, if M has codimension p, it is locally defined by p equations ρ j = 0, such that dρ j are linearly independent on M. Replacing ρ j by a jk ρ k =: ρ ′ j , for a suitable matrix of functions a jk , we may assume that n j := dρ j are orthonormal on M.
Then the currents of integration on M can be written The supercurrent associated to M is defined as The definition uses the forms n j that are only locally defined, but it is easily verified that a different choice n ′ j leads to the same supercurrent, since n j and n ′ j are related on M by an orthogonal transformation. If α is a superform on the ambient space, which acts on a superform or supercurrent by first contracting with n # and then wedging with F . Notice that F is an antiderivation. Then In a similar way, when p > 1 we let In the same way we get that Because of the following lemma, we can also write F = j n # j ⌊⊗F j , i. e. we can first wedge with F j and then contract. We prove this when p = 1. This is only to simplify the index notation; in the proof n j denotes the components of n, i. e. the partial derivatives of ρ with respect to x j . Then Since the norm of n is constant on M, this vanishes on M s .

MINIMAL SUBMANIFOLDS
We start with the following computational proposition.
Proof. We have, since dβ = 0,  (There seem to be different conventions as to the sign of the mean curvature vector. We follow here the convention in [10], so that the mean curvature of a sphere points outwards.) Applying Proposition 5.1 with p = m − 1 we get Since n j are linearily independent, this vanishes exactly when H vanishes, i. e. when M is a minimal manifold, so we have proved We are therefore led to the following It is clear that this property is conserved by regularisation, e.g. by convolution with an approximate identity. For a general manifold, if it happens that there is a vector field we say that V is a mean curvature vector for 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. In both these cases, V is uniquely determined, so we may speak of the mean curvature vector.
On a minimal supercurrent we can also define a Dirichlet form: if u is sufficiently smooth on R n . When T = [M] s is the supercurrent associated to an mdimensional manifold, this is precisely the standard Dirichlet form where d M u is the differential of u restricted to M, and |d M u| is the norm induced by the Euclidean metric on R n . If u has compact support, this equals It is therefore natural to define the Laplacian of u by With this definition, ∆ T u is a measure on R n . In some cases, e.g. when T is the supercurrent of a manifold, or a smooth strictly positive form, we can write where∆ T is a scalar valued Laplacian. In any case, we say that u is harmonic (subharmonic) on T if ∆ T u = 0 (or ∆ T u ≥ 0). Notice that, just as in the case of Kähler metrics, the Laplacian on a minimal supercurrent has no first order terms, so linear functions are harmonic. For minimal manifolds this is a well known property, cf [9].

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 ∧ β m−1 that is closed. This is taken care of by the following lemma. Then, if p is an integer, Proof. This is a direct computation and we will do it for p > 0, the case p = 0 being similar but simpler. First, Expanding by the binomial theorem we get The lemma follows from (4.1) and (4.2).
The reason we consider E p,δ instead of E p,0 is just that we want our functions to be smooth across zero. The main conclusion we draw from the lemma is that m). This follows from Proposition (2.1) since |x| δ is convex. We also remark that it follows from the proof of the proposition that Let 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 σ(r) := |x|<r T ∧ β m /m!.
In the computations below we first assume that T is smooth. Then, writing S = T ∧ β m−1 /m! σ(r) = since dS = 0. Since the integrand in the right hand side is nonnegative it follows that is (weakly) increasing. This holds for any δ > 0, so r −m σ(r) is also increasing. By approximation with smooth forms, this holds also for general minimal currents, so we have proved the following generalization of the monotonicity theorem for minimal manifolds (see [9]). When 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 Let us now look again at the formula We proved this for 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 δ → 0, so there is a subsequence of δ:s such that converges weakly to a measure, µ. Outside the origin, where |x| is smooth, this measure must be We also see that µ must have a point mass at the origin of size γ T (0), since the mass of µ in any ball centered at the origin with small radius r is Thus, every subsequence has the same limit

By Lemma 4.3 we also have
which can be interpreted as saying that the 'Laplacian' of E m−2,0 on T (the trace of dd # E m−2,0 ) equals a point mass at the origin of size γ 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 '. We shall next generalize the proof of Theorem 6.2 to a general domain, 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 ofD. Then the mass of T in D is (Here we skip the part of the argument where we approximate |x| by |x| δ and we write E m−2 for E m−2,0 .) By Stokes' theorem this equals since S is closed. By what we have just seen, I ≥ w(0)γ T (0). To see when II is positive we compute (for m > 2) It follows that II ≥ 0 if w ≥ |x| m in D and w is convex. (We shall see in the next section that the convexity assumption on w can be relaxed considerably.) In the proof we may of course assume that a = 0. For m > 2 the theorem follows immediately from the argument above, since we can always assume that w ≥ |x| m , replacing it if necessary by max(w, |x| m ). The case m = 2 is similar; we just have to replace the boundary integral in  Proof. We have |x − a| 2 = 1 + |a| 2 − 2a · x =: v on the boundary of the ball. Since v is convex (in fact linear), w := v m/2 is also convex and the corollary follows directly from the theorem.

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 ( [11], [16]) in the setting of minimal manifolds or minimal currents. We first define a locally integrable function u to be m-subharmonic if is a positive current for any (weakly) positive form α of bidimension (m, m). (In the terminology of complex analysis this amounts to saying that dd # u ∧ β m−1 be strongly positive, see [?].) Then (6.3) says that E p,δ is m-subharmonic if 0 ≤ p ≤ m − 2, and taking limits when δ → 0 the same thing holds for E p . Therefore, any potential Kernels like E m−2 on 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 [12]. 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 −∞ on F (and maybe elsewhere as well). By [12], any compact set K with C m−2 (K) = 0 is mpolar, and moreover there is a measure µ supported on K whose potential equals −∞ precisely on K. Therefore we get Proposition 7.1. Any compact set K of σ-finite (m − 2)-dimensional Hausdorff measure is mpolar and there is a potential u of a measure supported on K which equals −∞ 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 µ equal to surface measure. Theorem 7.2. Let T be a minimal supercurrent of bidimension (m, m), defined in B\K, where B is a ball and K and is compact in R n with sigma-finite (m − 2)-dimensional Hausdorff measure. Assume that T has finite mass Then the trivial extension of T ,T := χ B\K T , is a closed minimal supercurrent in B.
Proof. We follow almost verbatim the proof of Proposition 11.1 in [8]. Let u be a potential as in (4.7) with µ supported on K, which equals −∞ on K. Thus, u is smooth outside K and msubharmonic. Let χ(t) be an increasing continuous convex function, defined when t ≤ 0, with χ(0) = 1 and χ(t) = 0 for t ≤ −1. Put u k = χ(u/k) for k = 1, 2, .... Then u k are smooth, m-subharmonic, 0 ≤ u k < 1, and u k tend to 1 uniformly on compacts outside K.
Let θ be a smooth function with compact support in B. Then, by integration by parts, where C is a fixed constant independent of k. This implies that since u k is m-subharmonic. Let p(x) be a smooth function on R, such that p(x) = 1 if x > 1/2 and p(x) = 0 if x < 1/3, and put χ k = p(u k ). Then χ k tends to χ B\K and we get if ψ is a testform By our estimate on du k this tends to zero, so we are done.
One consequence of this is that Corollary 6.4 also holds for minimal manifolds with singularities. More precisely, by the theorem, the extensionT has a density also at points in K. It is easy to see, by the monotonicity theorem, that If T is associated with a minimal manifold of dimension m that has singularities at the set E the density is ω m (the volume of an m-dimensional unit ball) at all the regular points. Therefore it is at least ω 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 ?) We also note that the assumption that w be convex in Theorem 4.5 can be relaxed to msubharmonic, since we just need that  [10], [17]. As in these references, we say that a family M t of smooth submanifolds of R n moves by the mean curvature flow if there is a vector field H on the ambient space such that F t (M 0 ) = M t , where F t is the flow of − H, which restricts to the mean curvature vector field of M t on each M t .
Again we point out that we use here the same conventions about the sign of the mean curvature vector as in [10], which seems to be the opposite of the one in [17]. To make matters worse, the condition F t (M 0 ) = M t , means in terms of currents that F * −t ([M 0 ] s ) = [M t ] s , so since we will work with pullbacks, we are dealing with the flow G t of the vector field + H after all.
We first give a formula for how the volume form varies when a family of positive supercurrents T t of bidimension (m, m) moves under the flow of an arbitrary vector field, W , so that T t = G * t (T ), T = T 0 . By Cartan's formula (see the section on preliminaries), we have with Since dT ∧ β m = 0 and dT is of odd degree we get On the other hand , II is an exact form. If we assume that W has compact support in an open set D where T is defined, we see that i. e. T is minimal. Thus T is minimal if and only if its mass is stationary for all variations that vanish near the boundary, which gives another motivation for our terminology. We next assume that Hence, in this case we have (recall W # := W ⌋β) This is just the classical variation formula for the volume, [9]. We shall now see that when T = [M] s and W = H, we can also make the term II more precise; it is not just d-exact, but dd # -exact. In the statement of the theorem below we use our previous notation Proof. We apply the previous arguments to W = H and all that remains to prove is that As a consequence, if ρ is convex (or, more generally, m-subharmonic), then ρdσ t is decreasing.
It follows from the last part that if M lies in a convex open set, then M t will remain there, since we may choose ρ to be close to infinity outside the convex set.
Formula (8.2) 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 complete unrealistic.

GENERAL SUBMANIFOLDS AND THEIR RIEMANNIAN GEOMETRY
In this section we shall describe the Levi-Civita connection and Riemannian curvature of a submanifold M of R n in terms of the 'superstructure'.
Let M be a smooth submanifold of R n and let [M] s be its associated supercurrent. With the notation from section 3 we have Since F involves contraction with n # j , it vanishes on any form of bidegree (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, 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 ξ with values in the space of (p, 0)-forms in x. If V = V j ∂/∂x j is a tangent vector to M, the derivative of a in the direction V is It is not hard to verify that this is the Levi-Civita connection. 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,[19]).
Proposition 9.1. The curvature tensor R on M is given by the symmetric (2, 2) form in the sense that, if 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 ⌋a = n # j ⌋a = 0 for j = 1, ...p. It is obvious that canonical extensions exist locally since n j = dρ 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 ∧ [M] s .
If a is a canonical extension of a form on M s , then F a = F # a = 0, so To continue, we introduce the notation that if F = F ij dx i ∧ dξ j is a (1, 1) form (on R n s ) and a is (p, q) form, then F ∪ a = − F ij dξ j ∧ dx i ⌋a.
Notice that in the particular case when a has bidegree (1, 0) F ∪ a = −a⌋F. (1, 0)), then we have the commutator formula where a is any superform (canonical or not).
Proof. First we note that the commutator is a scalar operator, so that θ⌋]a if f is a function. Therefore we may assume that a = dx I ∧ dξ J . Then d # a = 0 and which proves the formula.
The lemma gives that (D # ) 2 a = − F j ∧ F j ∪ a, for a of any bidegree. Combined with the remark immediately before the lemma, this gives that (1, 0), which concludes the proof of the proposition. From Proposition 8.1 we see that integrals like where χ is a function on M are intrinsic, i. e. they do not depend on the embedding of M in 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.
We will now use the formalism of the previuos section to give a quick proof of Weyl's tube formula ( [20], [?]). The proof is not really different from the original proof, but the formalism helps to organise the formulas.
Let M be a compact submanifold of R n . We will consider T r (M), the tube around M of width r, which when M is without boundary is defined as A point v in the normal bundle is a vector, normal to to T p (M) where p = π(v), π being the projection from N(M) to M, and the diffeomorphism is When M is a compact manifold with boundary, we define T r (M) to be the image of ∆(M, r) under this map. Weyl's tube formula is the following theorem. for some constants c 2q that can be explicitly calculated, where p = n − m is the codimension of 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 R n , the tube volumes are the same.
For the proof, we have first where a = (a 1 , ...a p ) lies in R p . It clearly vanishes when m − l is odd, and when m − l = 2q it equals a constant times |a| 2q r 2q+p .
This follows since the integral is rotational invariant and homogenous of degree 2q in a, and homogenous of order 2q + p in r. Hence the integral in (9.2) equals ( a 2 j ) q r 2q+p This must also hold when a j are not real numbers but lie in any commutative algebra, like in our case when a j = F j are two-forms. Hence Inserting this into (9.1), the theorem follows.