Some compactness results by elliptic operators

Abstract In this paper, we get two compactness results for complete manifolds by applying a (sub-) elliptic second-order differential operator on distance functions. The first is an extension of a theorem of Galloway and gets an upper estimate for the diameter of the manifold and the second is an extension of a theorem of Ambrose.


Introduction
One of the most important and celebrated results in Riemannain geometry is the Myer's compactness theorem (Myers, 1941). This major theorem and its generalizations have many applications (Alvarez et al., 2015;Ambrose, 1957;Frankel & Galloway, 1981;Galloway, 1981). It states if M is a complete Riemannian manifold and its Ricci curvature is bounded bellow by ðn À 1Þa>0, then M is compact and its diameter satisfies diamðMÞ � ffi ffi π p a , also by the same argument for the universal covering space, one can conclude that M has finite first fundamental group. We recall two important generalizations of this theorem. The first is about Galloway's theorem as follows.
Theorem 1.1. (Galloway, 1981) (Galloway)Let M be a complete Riemannian manifold and for any unit vector field X, one has RicðX; XÞ � a þ Ñφ; X h i; where a is a positive constant and φ is any smooth function satisfying φ j j � c. Then M is compact and its diameter is bounded from above by Shahroud Azami ABOUT THE AUTHOR The author received his PhD from the Amirkabir University of Technology. He is working as professor at department of mathematics, faculty of sciences, Imam Khomeini international university, Qazvin, Iran. He published a number of research articles in international journals. He guided many postgraduate students. His research area is differential geometry.

PUBLIC INTEREST STATEMENT
The compactness theorem by Myer's and volume comparison theorem by Bishop-Gromov are essential tools in differential geometry and analysis on manifolds. In this paper, by using aelliptic second-order differential operator on distance functions, we give two compactness results for complete manifolds.
Both theorems above have nice applications in relativistic cosmology (see (Alvarez et al., 2015;Ambrose, 1957;Frankel & Galloway, 1981;Galloway, 1981)). Myer's theorem has been generalized in many ways, for example, Bakry and Qian applied the elliptic operator of the form Δ þ X, (where X is some vector field) to the distance function and get some estimate of the diameter of the manifold under some curvature-dimension inequality (Bakry & Qian, 2005 (Cavalcante et al., 2015) Let M be a complete Riemannain weighted manifold, V be a smooth vector field, and for any unit vector field X on M one has, where k and c are positive constant and φ is a smooth function with φ j j � c. Then M is compact and its diameter satisfies, diamðMÞ � π ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ðn À 1Þc p b ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ðn À 1Þc p þ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi b 2 ðn À 1Þc þ ðn À 1Þ þ 4k where Ric k f ¼ Ric þ Hessf À 1 k df � df and k 2 ð0; 1Þ. Then M is compact.
In this paper, we generalize Theorems 1.1 and 1.2 by means of some kind of (sub-)elliptic operators and lower bound of the extended Ricci tensor RicðX; AXÞ À 1 2 L V g ð Þ X; X ð Þ as follows, the first is an extension of Galloway's theorem.
Theorem 1.6. Let M be a complete Riemannian manifold, A a ð1; 1Þ-self adjoint tensor field, V 2 XðMÞ a smooth vector field and φ a smooth function with φ j j � K 0 . Fixed p 2 M and define rðxÞ ¼ distðp; xÞ. Assume H>0 be some constant and the following conditions are satisfied, a) for some constant H>0 and any unit vector field X 2 XðMÞ we have, where GðtÞ and f ðtÞ are defined in Lemma 3.1, ( for some constants K 2 ; K 3 ; K 5 ; K 6 ; K 7 . Then M is compact, its fundamental group is finite and its diameter satisfies, diamðMÞ � π ffi ffi ffiffi H p þ 1 δ n ðn À 1ÞH The second result is an extension of theorem of Ambrose.

Preliminaries
In this section, we present the preliminaries. Throughout the paper M ¼ ðM; h; iÞ is a complete Riemannian manifold. First, we give some definitions.
Definition 2.1. A self-adjoint operator A on M is a 1; 1 ð Þ-tensor field with the following property, "X; Y 2 XðMÞ; hAX; Yi ¼ hX; AYi: Now we define bounded operator A as follows.
By the following definition, we give some notations about second-order differential operator L on a manifold M with L1 ¼ 0. In fact a second-order differential L with L1 ¼ 0 can be written as Definition 2.4. Let A be a ð1; 1Þ-tensor field on M. Define T A as, It is clear that T A is a (2,1) tensor field.
Example 2.5. If A is the shape operator of a hypersurface � n � M nþ1 then where � R is the curvature tensor of M and N is a unit normal vector field on � n � M nþ1 .
We recall the following extended Bochner formula from (Alencar et al., 2015;Gomes & Miranda, 2018) to prove Theorem 2.10 which is the main tools to get the compactness results.
Proposition 2.6. Let A be a self-adjoint operator on M, then, where Ric A is defined in (Fatemi & Azami, 2018).
The term Δ ÑuA u in (2.1) is very complicated and depends on the algebraic and analytic properties of the tensor field A. So we try to simplify it to get the better estimates. First, by the following Lemmas, we show some relations about the second covariant derivative of the tensor field A. A be a (1,1)-self-adjoint tensor field on M and X; Y; Z 2 XðMÞ, then

Lemma 2.7. Let
Proof. For part (a) we have, Similarly, For part (b), by definition of T, we have Lemma 2.8. Let A be a ð1; 1Þ À self-adjoint tensor field on M, then where Ñ � is adjoint of Ñ and Proof. For simplicity let e i f g be an orthonormal local frame field in a normal neighborhood of p such that with Ñ e i e j ¼ 0 at p. At p Lemma 2.7 implies, So by Lemma 2.7, part (a) we have Now, we ready to simplify the term Δ ÑuA u in (1).

Proposition 2.9. Let A be a (1,1)-self-adjoint tensor field on M and u 2 C 1 ðMÞ, then
Proof. Let A be a ð1; 1Þ-tensor field, then In other words, But, So, Finally the result concludes by Lemma 2.8.
Here is another extension of Bochner-formula, which we use it as one of the mail tools to get the compactness results.

Extended Laplacian comparison theorem
In this section we shall extend the mean curvature comparison theorem by some ( where G : R ! R be a smooth function. Proof. By sec rad M � À GðrðxÞÞ we have the following estimate for HessrðX; XÞ :¼ Ñ X Ñr; X h i; (see (Pigola et al., 2008) where the radial sectional curvature satisfies sec rad � À G and f be the solution of differential equation (3.1), then f A can be the solution of the following differential inequality, Here we get an extension of mean curvature comparison theorem.
Theorem 3.4 (Extended mean curvature comparison) Let M be a complete Riemannian manifold, A a ð1; 1Þ-self-adjoint tensor field, V 2 XðMÞ a smooth vector field and φ a smooth function with φ j j � K 0 . Fixed p 2 M and define rðxÞ ¼ distðp; xÞ. Assume H>0 be some constant and the following conditions are satisfied, a) for any unit vector field X 2 XðMÞ we have, Proof of Theorem 3.4. We are inspired by the proof of (Fatemi & Azami, 2018). By Lemma 3.1 and Theorem 2.10, we get the following differential inequality, 0 � Δ A r ð Þ 2 ðn À 1Þδ n þ @r:Δ A;V r À @r:@r:trðAÞ þ @r: divA; @r h i À K 1 ðrÞ f 0 ðrÞ f ðrÞ À @r:@r:f A þ Ric @r; A@r ð Þ À 1 2 L V g ð Þ @r; @r ð Þ: Let γðtÞ be a minimal geodesic through the point x 0 .