UDC 513.836

MATHEMATICS

A. T. FOMENKO

THE MULTIDIMENSIONAL PLATEAU PROBLEM AND SINGULAR POINTS OF MINIMAL COMPACTS

(Presented by Academician P. S. Aleksandrov on 21 X 1969)

1. Let \((\mathfrak M^n, A)\) be a compact pair, where \(\mathfrak M^n\) is a compact, closed Riemannian manifold. In note \((^4)\) the classes \(O^k(A,\mathfrak L,\mathfrak L')\), \(R^k(A)\), \(N^k(A)\) were introduced and the corresponding existence theorems for minimal compacts were formulated. It was also noted there that a special case of the class \(O^k(A,\mathfrak L,\mathfrak L')\) is the class
   \[ O^k(A,\mathfrak L,0)=\mathfrak G^k(A,\mathfrak L), \]
   studied by Reifenberg \((^1)\) and Morrey \((^2)\). In the case when the coefficient group \(\mathfrak G=U=\mathbf R^1(\bmod 1)\), the class \(N^k(A)\) coincides with the class \(\mathfrak G^*\) considered by Reifenberg in \((^1)\).

In the present note a new class \(P^k(A,\mathfrak L,\mathfrak L')\) is introduced, in which the multidimensional Plateau problem is posed and solved. In addition, the note indicates a connection between the homological structure of minimal compacts in the classes \(O^k(A,\mathfrak L,\mathfrak L')\) and properties of the set \(Z\) of their singular points.

1. By \(H_*(X;\mathfrak G)\) we shall denote the Čech–Aleksandrov homology of the space \(X\) with coefficients in the Abelian group \(\mathfrak G\). In what follows, by \(\mathfrak G_C\) we denote the category of compact Abelian groups, and by \(\mathfrak G_F\) the category of linear vector spaces over some field \(F\). Consider a compact closed Riemannian manifold \(\mathfrak M^n\), and let
   \[ A\subset \mathfrak M^n \]
   be an arbitrary but fixed compact set. Let
   \[ \mathfrak L\subset H_{k-1}(A;\mathfrak G) \]
   be an arbitrary subgroup, \(\mathfrak G\in\mathfrak G_C\) or \(\mathfrak G\in\mathfrak G_F\). Consider the class \(O^k(A,\mathfrak L,0)\) (for the definition see \((^4)\)) and suppose that it is nonempty if the subgroup \(\mathfrak L\) is nontrivial. Let \(\mathfrak L'\) be an arbitrary subgroup in the group of relative homologies \(H_k(\mathfrak M^n,A;\mathfrak G)\), and suppose that at least one of the subgroups \(\mathfrak L\) and \(\mathfrak L'\) is nontrivial. If \(\mathfrak L=0\), then the class \(O^k(A,0,0)\), obviously, coincides with the class of all compacts such that \(X\subset\mathfrak M^n\) and \(A\subset X\).

Definition 1. Consider the triple \((A,\mathfrak L,\mathfrak L')\). We shall say that a compact \(X\subset\mathfrak M^n\) belongs to the class \(P^k(A,\mathfrak L,\mathfrak L')\) if
\[ X\in O^k(A,\mathfrak L,0) \]
and
\[ \omega_*(H_k(X,A;\mathfrak G))\supset \mathfrak L', \]
where
\[ \omega_*:H_k(X,A;\mathfrak G)\to H_k(\mathfrak M^n,A;\mathfrak G) \]
is the homomorphism induced by the inclusion
\[ \omega:(X,A)\to(\mathfrak M^n,A). \]

For \(\mathfrak L=0\) we obtain compact pairs \((X,A)\) realizing relative cycles in the group \(H_k(\mathfrak M^n,A;\mathfrak G)\). It is clear that
\[ P^k(\varnothing,0,\mathfrak L')=O^k(\varnothing,0,\mathfrak L'); \qquad P^k(A,\mathfrak L,0)=O^k(A,\mathfrak L,0). \]
If one puts \(\mathfrak L=0\) and \(\mathfrak L'=\{\sigma\}\), a subgroup in \(H_k(\mathfrak M^n,A;\mathfrak G)\) generated by the element \(\sigma\), then one obtains the class \(P^k(A,0,\{\sigma\})\), studied by Almgren in \((^3)\) for the case of \(k\)-spanning compacts and finitely generated coefficient groups. Let us note that the relation
\[ \mathfrak L'\subset \operatorname{Im}\omega_* \]
is equivalent to the condition that
\[ \mathfrak L'\subset \operatorname{Ker}\varphi_*, \]
where
\[ \varphi_*:H_k(\mathfrak M^n,A;\mathfrak G)\to H_k(\mathfrak M^n,X;\mathfrak G). \]

The technique used in the proof of the existence theorems in the classes \(O^k(A,\mathfrak L,\mathfrak L')\), \(N^k(A)\), \(R^k(A)\) (see \((^4)\)) makes it possible to obtain an existence theorem also in the class \(P^k(A,\mathfrak L,\mathfrak L')\).

1. Theorem 1. Let \(\mathfrak M^n\) be a compact, closed Riemannian manifold of class \(C^p\), where \(p\ge 4\); let \(A\subset\mathfrak M^n\) be an arbitrary compact set; let \(\mathfrak G\in\mathfrak G_C\) or \(\mathfrak G_F\); let \(k\) be an integer and \(k\ge 3\); let \(\mathfrak L\) and \(\mathfrak L'\) be subgroups in the groups \(H_{k-1}(A;\mathfrak G)\) and \(H_k(\mathfrak M^n,A;\mathfrak G)\), respectively, where at least one of them is nontrivial.

Suppose that \(P^k(A,\mathfrak L,\mathfrak L')\ne \varnothing\) and \(\mu(A,\mathfrak L,\mathfrak L')<\infty\), where
\[ \mu(A,\mathfrak L,\mathfrak L')=\inf \Lambda^k(X\setminus A),\quad X\in P^k(A,\mathfrak L,\mathfrak L'). \]
Then there exists a compact set \(X_0\in P^k(A,\mathfrak L,\mathfrak L')\) such that \(\Lambda^k(X_0\setminus A)=\mu(A,\mathfrak L,\mathfrak L')\), and every point \(x\in (X_0\setminus A)\setminus Z\), where \(\Lambda^k(Z)=0\), possesses in \(X_0\) a neighborhood homeomorphic to the \(k\)-dimensional disk \(D^k\). Moreover, if \(\mathfrak M^n\in C^4\), then these \(k\)-disks may be assumed to belong to the class \(C_\mu^3\) for any \(0<\mu<1\); if \(\mathfrak M^n\in C_\mu^p\) for some \(p\ge 4\) and some \(\mu\), \(0<\mu<1\), then the \(k\)-disks may be assumed to belong to the class \(C_\mu^p\); finally, if \(\mathfrak M^n\in C^\infty\) or is analytic, then the \(k\)-disks may be assumed to belong to the class \(C^\infty\) or to be analytic, respectively. In addition, \((X_0\setminus A)\setminus Z\) is a submanifold in \(\mathfrak M^n\) of the corresponding smoothness.

1. We proceed to the study of singular points of minimal compact sets \(X_0\in O^k(A,\mathfrak L,\mathfrak L')\). Consider in the manifold \(\mathfrak M^n\) the open submanifold \((X_0\setminus A)\setminus Z\) (see the main theorem in \((^4)\)), which, generally speaking, is not connected. Let
   \[ (X_0\setminus A)\setminus Z=\bigcup_i \Pi_i, \]
   where \(\Pi_i\) denotes the components of linear connectedness, which are differentiable submanifolds in \(\mathfrak M^n\). If the submanifold \((X_0\setminus A)\setminus Z\) consists of a finite number of linearly connected components, we shall denote their number by \(N\).

Definition 2. A component \(\Pi_i\) is called \(\mathfrak G\)-orientable if
\[ H_k(\overline{\Pi}_i,\partial\Pi_i;\mathfrak G)\ne 0, \]
where the closure \(\overline{\Pi}_i\) is taken in \(\mathfrak M^n\), \(\partial\Pi_i=\overline{\Pi}_i\setminus \Pi_i\), \(k=\dim X_0\). Let us note that here the group \(\mathfrak G\) is precisely the group which characterizes the class \(O^k(A,\mathfrak L,\mathfrak L')\).

Since
\[ H_k(\overline{\Pi}_i,\partial\Pi_i;\mathfrak G)\cong H_k(\overline{\Pi}_i/\partial\Pi_i;\mathfrak G), \]
we are in fact speaking of the \(\mathfrak G\)-orientability of the compactification of the manifold \(\Pi_i\) by means of the point \(\omega\). The number of \(\mathfrak G\)-orientable components of the manifold \((X_0\setminus A)\setminus Z\) will be denoted by \(N_0\).

Theorem 2. Let \(\mathfrak M^n\) be a compact, closed Riemannian manifold, \(\mathfrak M^n\in C^4\), \(\mathfrak G=Z_p\), \(p\ne 0\), \(p\) prime, \(A\subset \mathfrak M^n\), \(\Lambda^k(A)=0\), and let \(X_0\in O^k(A,\mathfrak L,\mathfrak L')\) be a minimal compact set, and suppose \(N<\infty\). Then:

I. If \(p=2\), then
\[ \dim \mathfrak L+\dim \mathfrak L'\le N\le \dim H_k(\mathfrak M^n;Z_2)+\dim H_{k-1}(A\cup Z;Z_2). \]

II. If \(p\ne 2\), then
\[ \dim \mathfrak L+\dim \mathfrak L'\le N_0\le \dim H_k(\mathfrak M^n;Z_p)+\dim H_{k-1}(A\cup Z;Z_p). \]

Corollary 1. Suppose that all the assumptions of Theorem 2 are fulfilled, and suppose additionally that \(\mathfrak L'=0\) and \(\Lambda^{k-1}(Z)=0\). Then for \(p=2\)
\[ N=\dim H_{k-1}(A;Z_2); \]
for \(p\ne 2\)
\[ N_0=\dim H_{k-1}(A;Z_p). \]

We emphasize that the numbers \(N\) and \(N_0\) refer to different minimal compact sets corresponding to different coefficient groups.

1. We shall now consider the class \(O^k(\varnothing,0,\mathfrak L')\), \(\mathfrak L'\ne 0\); a minimal compact set from this class will be denoted by \(Y_0^m\) (the letter \(m\) indicates minimality). Of all minimal compact sets \(X_0\in O^k(A,\mathfrak L,\mathfrak L')\), the compact sets \(Y_0^m\) are apparently arranged most simply, since they are not obliged to span any boundary \(A\subset \mathfrak M^n\). By \(Z\), as usual, we denote the set of singular points of the compact set \(Y_0^m\); it is known (see \((^4)\)) that always \(\Lambda^k(Z)=0\). It turns out that the presence in the compact set \(Y_0^m\) of certain homological properties entails the appearance of a set \(Z\) of singular points of the greatest possible (in the sense of dimension) positive measure.

Theorem 3. Let \(\mathfrak M^n\) be a compact, closed Riemannian manifold, \(\mathfrak M^n\in C^4\), \(\mathfrak G=Z_p\), \(p\ne 0,2\), \(p\) prime; \(Y_0^m\in O^k(\varnothing,0,\mathfrak L')\), \(\mathfrak L'\ne 0\). Then, if \(N<\infty\) and
\[ \dim H_k(Y_0^m;Z_2)\ne \dim H_k(Y_0^m;Z_p), \]
then \(\Lambda^{k-1}(Z)>0\), and at least one of the groups
\[ H_{k-1}(Z;Z_2),\qquad H_{k-1}(Z;Z_p) \]
is nontrivial. Moreover, all components \(\Pi_i\) are \(Z_p\)-orientable.

Definition 3. We shall say that a subgroup \(\mathfrak L'\subset H_k(\mathfrak M^n;\mathfrak G)\) admits an exact minimal realization if
\[ \mathfrak L'\cong H_k(Y_0^m;\mathfrak G), \]
where \(Y_0^m\in O^k(\varnothing,0,\mathfrak L')\).

In the group $H_k(\mathfrak M^n; Z_p)$ one can always choose a basis consisting of elements $e_1, e_2,\ldots,e_r$, $r=\dim H_k(\mathfrak M^n; Z_p)$, admitting an exact minimal realization. Let $Y_{0j}^m \in O^k(\varnothing,0,\{e_j\})$ be the minimal carriers of the one-dimensional subgroups $\{e_j\}$, $1\le j\le r$.

Corollary 2. Let $\mathfrak G=Z_p$, $p\ne0,2$, $p$ prime, and let $N<\infty$. Then, if $\dim H_k(Y_{0j}^m;Z_p)\ne1$, then $\Lambda^{k-1}(Z)>0$ and at least one of the groups $H_{k-1}(Z;Z_2)$, $H_{k-1}(Z;Z_p)$ is nontrivial.

Theorem 3 and Corollary 2 admit a transparent geometric interpretation. Roughly speaking, if an integral cycle is realized by means of a closed orientable manifold $V^k$, then
$\dim H_k(V^k;Z_p)=\dim H_k(V^k;Z_2)$ and the set of singular points is empty. On the other hand, cycles of order $p$ are arranged as CW-complexes $V^{k-1}\cup_f W^k$, where $\partial W^k=V^{k-1}$, and $f$ is a mapping $V^{k-1}\to V^{k-1}$ of degree $p$ (the manifold $W^k$ is attached to $V^{k-1}$ by this mapping), which leads to the inequality
$1=\dim H_k(Y_0^m;Z_p)\ne\dim H_k(Y_0^m;Z_2)=0$ and to the appearance of a set $Z$ of singular points ($Z=V^{k-1}$) for $p\ne2$, and moreover $\Lambda^{k-1}(Z)>0$. It is also clear that for $p=2$ the set $Z$ may be empty; for example, $Z=\varnothing$ for the projective space $RP^k$, where $V^{k-1}=RP^{k-1}$, $W^k=D^k$.

1. Consider the special case when the compact set $A$ is a $(k-1)$-dimensional sphere $S^{k-1}$ embedded in $\mathfrak M^n$, and assume that the embedding is such that the class $N^k(A)$ is nonempty (see the definition in $(^4)$). As the coefficient group $\mathfrak G$ take the group $U=R^1(\bmod 1)$. Such a fixed topological structure of the boundary allows one to obtain certain information of purely metric character concerning the structure of the minimal compact set $X_0\in O^k(A,\mathfrak L,\mathfrak L')$. We note that with each embedding $S^{k-1}\to\mathfrak M^n$ two numbers are associated: $d'$ and $\tilde d$, where
   $d'=\inf\Lambda^k(X\setminus A)$, $X\in R^k(A)$; $\tilde d=\inf\Lambda^k(X\setminus A)$, $X\in N^k(A)$ (see $(^4)$); assume that $\tilde d<\infty$. Since in our case $\Lambda^k(A)=0$, instead of the class $R^k(A)$ one may consider the broader class $R^k(\varnothing)\supset R^k(A)$, with
   $\inf\Lambda^k(X\setminus A)=d'$, where $X\in R^k(\varnothing)$. This means that the number $d'$ is in fact determined only by the manifold $\mathfrak M^n$, namely: it is the Hausdorff measure of the “least” $k$-dimensional cycle in $\mathfrak M^n$. The number $\tilde d$ is the Hausdorff measure of the “least” film spanning a nontrivial subgroup in the group $H_{k-1}(S^{k-1};U)$. Consider the decomposition of the submanifold $(X_0\setminus A)\setminus Z$ into connected differentiable submanifolds $\Pi_i$ (the number of components may also be infinite). The question arises: how “massive,” in the sense of the measure $\Lambda^k$, can the individual components $\Pi_i$ be.

Theorem 4. Let $\mathfrak M^n$ contain the compact set $A=S^{k-1}$, $X_0\in O^k(A,\mathfrak L,\mathfrak L')$, where $\mathfrak L\ne0$, $\mathfrak L'=U$; $d=\Lambda^k(X_0\setminus A)$; $\Delta=\max(d-d',\,d-\tilde d)$. Then $\Delta>0$ and
\[ \sup_i \Lambda^k(\Pi_i)\le \Delta . \]

Remark. In the general case the estimate obtained in Theorem 4 cannot be improved: it is easy to indicate an example in which the value is attained for some $i=i_0$.

1. In Reifenberg’s paper $(^1)$ an existence theorem was proved for a minimal compact set in the class $\mathfrak G^*=N^k(S^{k-1})$, and, what is especially interesting, the compact sets $X\in\mathfrak G^*$ were described in terms of retraction mappings. Consider a compact pair $(X,A)$, where $A=S^{k-1}$, coefficient group $\mathfrak G=U$. Reifenberg introduced two definitions of a boundary (see $(^1)$). We shall say that $A=\partial^R(X)$ if there does not exist a $(k-1)$-dimensional set $A^*\subset X$, $A^*\supset A$, such that $A^*=\operatorname{retr}X$ ($A^*=\operatorname{retr}X$ if there exists a continuous mapping $f:X\to A^*$ such that $f(A^*)\equiv A^*$). We shall say that $A=\partial^{\mathfrak L A}(X)$ if $\mathfrak L=b(X,A)\ne0$ (see the definition in $(^1)$). From Hopf’s extension theorem it follows that these two definitions are equivalent. In $(^1)$ Reifenberg put forward the hypothesis that this equivalence also holds in the case when $A=\mathfrak M^{k-1}$ is an arbitrary compact closed orientable manifold. The following example shows that this hypothesis is false. Put $A=S^1\times S^{k-2}$;

then \(\pi_{k-1}(A)=Z_2\) for \(k \geq 5\). Let \(a \in [a] \in Z_2\) be a continuous mapping \(S^{k-1}\to A\) corresponding to the generator \([a]\). Put \(X=A\cup_a D^k\), attaching to \(A\) a \(k\)-dimensional disk \(D^k\) by the mapping \(a\). It is clear that \(A=\partial^R(X)\), but \(A\ne \partial^{\mathfrak L A}(A)\) for any nontrivial subgroup \(\mathfrak L\).

I take this opportunity to express my gratitude to Prof. P. K. Rashevskii for his constant attention to this work.

CITED LITERATURE

\(^{1}\) E. R. Reifenberg, Acta Math., 104, No. 1 (1960).
\(^{2}\) Ch. B. Morrey, Multiple Integrals in the Calculus of Variations, Berlin, 1966.
\(^{3}\) F. J. Almgren, Ann. Math., Ser. 2, 87, No. 2 (1968).
\(^{4}\) A. T. Fomenko, DAN, 187, No. 4 (1969).