ON MINIMAL HYPERSURFACES OF NONNEGATIVELY RICCI CURVED MANIFOLDS

We consider a complete open riemannian manifold M of nonnegative Ricci curvature and a rectifiable hypersurface ∑ in M which satisfies some local minimizing property. We prove a structure theorem for M and a regularity theorem for ∑. More precisely, a covering space of M is shown to split off a compact domain and ∑ is shown to be a smooth totally geodesic submanifold. This generalizes a theorem due to Kasue and Meyer.

In this paper, by a compact hypersurface E, we mean an integral current in M of Hausdorff dimension d 1 It is well known that the support, spt E, of such a E is almost everywhere (with respect to Hausdorff measure) the union of a finite number of Lipschitz images in M of compact d dimensional orientable riemannlan manifolds.
In many ways, rectifiable currents are the correct setting for the theory of minimal submanifolds, because they are closed under taking weak limits.We refer to the books of Federer  [3], Lawson [4], and Morgan [5].We say that E is without boundary if the d 2 dimensional current aE is identically O.
For an open set U , let E]U denote the localization of E to the closure U of In this paper, we choose to call a current E locally minimizing if, for each x e , there exists some neighbor- hood U of x in such that the varifold associated to E[U is abso- lutely mass minimizing for the variational problem supported on Although this condition is stronger than that of being merely statio- nary and is not a closed condition under the weak topology, it includes the solutions to most pertinent variational problems.
Notice that It is weaker than the notion of locally minimizing as described by Lawson in [4] where E is required to be absolutely minimizing with respect to all variations with compact support.In particular, our condition is satisfied by all weakly embedded smooth minimal submanifolds.
Technically, our definition has the advantage of satisfying locally the the requirements for the regularity heorems of Almgren [6].MAIN THEOREM.Suppose ha here exists in a locally minimizing compac hypersurface E wihou boundary in he sense described above.
Then, for some covering space p'M M, there exists a smooth compact orienable d-1 dimensional riemannlan manifold T and possibly empty compac reglon O M such that M \ O splits iometri- cally as either the riemannian product T x or T x 0,) and spt p*E consists of disjoin smoothly embedded copies of Notice that in particular, our Main theorem implies the follow- ing regularity theorem.
COROLLARY.A locally minimizing compac hypersurface Ithou boundary in a complete noncompac manifold M of nonnegative Rlccl curvature must be regular everywhere and totally geodesic.
In this sense, we point out its relation to the works of Ander- son [7] and Anderson and Rodriguez [8].
Of course, under our Main theorem, the Ricci curvature, in fact the sectional curvature, of M in the direction of the linear factor is 0. Therefore, we obtain, COROLLARY.Suppose that M is a complete riemannian manifold of positive Ricci curvature.
If there exists a minimal compact hyper- surface E in M without boundary, then M itself must be compact.
This generalizes a part of our result in [9].
Let M be a complete connected noncompact riemannian manifold and let d be its dimension.
We recall from D. Gromoll and W. Meyer [10] that a ray in M is a geodesic c' [O,)  M such that every segment c[[0, b] In [11], J. Cheeger and Gromoll prove that b is always conti- c nuous and that if M has nonnegative Ricci curvature, then b is c superharmonic.
Moreover, Galloway [12] proves that T is any piece of smooth minimal hypersurface of then b T obeys the Strong minimum c 'principle; i.e., If U c T is a relatively open connected set of T and b attains a local minimum value in U, then b is constant throughout U.
Actually, the result, Lemma (2.4) in [12], is stated for the case M is a Lorentzian spacetime with positive timelike Ricci curvature, but the same proof goes through for our case.Cf. also a related result of Galloway and Rodriguez [13].
LEMMA I. Let A c M be any compact set.Then, for each end H of M there exists a ray c in M such that c(O) e A c(s) e g for large s, and b is nonnegative on A. c Here, by c(s) E for large s, we mean that for each component E of the complement of the filtration defining the end there is a t > 0 such that c(s) e g for all s > t PROOF.Take a sequence of points qj in g such that p(A, qj) > 1/i Since A is assumed to be compact, for each i, there is a unit-speed geodesic segment c [0,1 l] M realizing the distance between A and q Let TM M be the unit tangent sphere bundle of Then, c(0) l -(A) while the latter is a compact space and so the sequence of tangent vectors (0) has some sub- sequence converging to a vector u -l{A) Then, c(s) exp su is a ray with c(O) A and c(s) for large s.
Moreover, if we define a sequence of functions g'M gi (p) p(p, ql) 1 then, g converge pointwise to the Busemann function bc Since gilA is nonnegative, we obtain the lemma.
LEMMA 2. Lt c be a ray constructed for a compact set A as in Lemma I.Then, q c(O) is the closest point in A to any point on 576 Y. ITOKAWA fhe ray c.

PROOF. Let x ()
Assume that there is a point y A which is closer to x than q.Then, p(y,x) for some > 0 Hence, for all s > t p(y,c(s) < p(y,x) + p(x,c(s) t-e + s-t s-e It follows that b (y) lira p(y,c(s)) s e which contradicts Lemma 1.We mention that Shioya [14] made the same observations as our two preceding lemmas but used them for a different purpose.Now, let E be a locally minimizing compact hypersurface in M and let A be the support of E. Construct a ray c associated to A as in Lem=a 1.We recall an observation of M. Gromov [15] that for any point x A the point y A which is closest to x is a regular point of E in the sense that there is a neighborhood U of q in M so that spt {El is a smoothly embedded image of a piece of d-1 dimensional smooth manifold.Therefore, by virtue of Lemma 2, we see, LEMMA 3. q := c(0) is a regular point of .
Let S be the connected component of q in spt and W the smoothly embedded piece of d I manifold defining IU in a neighbor- hood U of q.We may assume that W S. We can then apply the ]emma of Galloway and Rodrlguez to conclude that b IW obeys the Strong minimal principle.
But, b [S k 0 and b (q) 0 SO q is a minimum In particular, S is regular everywhere.
Moreover, by the techniques of Kasue [I] and Meyer [2], this induces a local splitting of M; i.e., M contains a region R in turn containing Wn R as part of its boundary and R splits isometrically as the product (Wo R) [0,) Now, we recall that for purely topological reasons, there is a covering space p:M M of M so that p-I (S) decomposes into a finite number of connected components each of which, say divides M itself into two connected regions with T as its common boundary.
Then, it is easily seen that, passing to the splitting of p'(R) extends to the global splitting T [0,) Now, let P be the region in isometric to T [0,) and let O := M\ P. If the closure of Q is not compact, then Q contains an end of We can construct a ray as in Lemma 1 for the compact set T and argue as before to conclude that Q also splits as T (-,0] Otherwise, the closure of O has to compact.This proves the first half of our Main theorem. 0 < b, is length minimizing.The function b "M R defined by b (p) "=, im p(p, c(s)) s c where p is the distance function in M is called the Busemann function associated to c.
in S.Moreover, q is an interior point of so we C conclude that b 0 on But since b is continuous and S is c c connected, b is in fact identically 0 on S.