Heat kernel on connected sums of Riemannian manifolds

Alexander Grigor’yan ^{∗} Department of Mathematics

180 Queen’s Gate Huxley Building Imperial College London SW7 2BZ United Kingdom a.grigoryan@ic.ac.uk

Laurent Saloff-Coste ^{†} CNRS, Toulouse, France and Department of Mathematics Malott Hall Cornell University Ithaca, NY 14853-4201 United States lsc@math.cornell.edu

May 1999

1 Introduction

This note is about the heat kernel on a connected sum M of non-compact manifolds M_{1}, M_{2}, ..., M_{k} assuming that one knows enough about the heat kernels for each M_{i} individually (which is the case when M_{i} are complete manifolds of non-negative Ricci curvature). We announce here matching uniform upper and lower bounds for the heat kernel on such manifolds M. The proofs will be given elsewhere.

For an arbitrary Riemannian manifold M, denote by Δ the Laplace operator of the Riemannian metric of M and by p(t, x, y) the heat kernel, that is, the smallest positive fundamental solution to the heat equation u_{t} = Δu on ℝ_{+} × M (here x, y ∈ M and t > 0). Alternatively, p(t, x, y) can be defined as the kernel of the semigroup exp (tΔ). Yet another definition of p(t, x, y) is that it is the density of the transition probability of the Brownian motion on M generated by the operator Δ.

In ℝ^{n}, the heat kernel is given by the classical formula

p(t, x, y) = \frac{1}{(4πt)^{n/2}} exp (−\frac{|x − y|^{2}}{4t} ).  (1)

∗Research supported by EPSRC Fellowship B/94/AF/1782 (England)

†Research supported by NSF grant DMS 9802855

On a general manifold M, one may hope to obtain some estimates of the heat kernel related to the geometry of M. In the past two decades, this question has received considerable attention from many authors. This was partially motivated by various important applications of heat kernel bounds although the question of understanding the interplay between the heat kernel and the geometry is interesting by itself. The celebrated theorem of Li and Yau [16] says that if M is a geodesically complete manifold with non-negative Ricci curvature then

\frac{c}{V (x, \sqrt{t})} exp (−\frac{d^{2}(x, y)}{ct}) ≤ p(t, x, y) ≤ \frac{C}{V (x, \sqrt{t})} exp (−\frac{d^{2}(x, y)}{Ct}) . (2)

Here V (x, r) is the Riemannian volume of the geodesic ball B(x, r) of radius r centered at x ∈ M; d(x, y) is the geodesic distance between the points x, y ∈ M; C, c are positive constants. There are many other settings when certain heat kernel bounds are known - see for example [1], [3], [4], [7], [8], [11], [12], [18], [17].

In most known estimates the heat kernel is controled by two multiplicative terms: a Gaussian factor exp(−\frac{d^{2}(x,y)}{Ct}) and a long time decay term which is some function of time t as in (1) and (2). The reason is that in most situations considered previously the manifold M possesses a certain degree of homogeneity so that p(t, x, y) depends on x and y mainly via the distance d(x, y). At the present time, there are no methods yielding satisfactory heat kernel bounds for highly inhomogeneous manifolds having substantially different geometric properties in different parts. The manifolds with ends considered in this note can be regarded as the simplest case of such non-homogeneous manifolds.

The main results are Theorems 1 and 4. Since the results are not simple, we first explain them for connected sums of Euclidean spaces (Section 2). The general case is considered in Sections 3 and 4.

2 Connected sums of Euclidean spaces

Let M_{1}, M_{2},.., M_{k} be a finite family of geodesically complete non-compact Riemannian manifolds. We say that a manifold M is a connected sum of M_{i}’s and write

M = M_{1}#M_{2}#...#M_{k}

if, for some compact K ⊂ M (which will be called the central part of M), the exterior M \ K is a disjoint union of open sets E_{1}, E_{2},..., such that each E_{i} is isometric to M_{i} \ K_{i}, for some compact K_{i} ⊂ M_{i} (in fact, we will identify E_{i} and M_{i} \ K_{i}). In other words, in order to obtain M, one cuts out holes K_{i} in each M_{i} and glues together the rest of M_{i}, that is E_{i}, over some compact K. Of course, a connected sum is not uniquely defined. When writing M_{1}#M_{2}#..., we refer to any manifold which can be obtained in this way.

Another point of view is to regard M as a manifold with ends E_{i} assuming that each end E_{i} can be closed to a complete manifold M_{i}. Sometimes it is convenient to say that M_{i}’s are the ends of M.

The connected sum ℝ^{n}#ℝ^{n} was considered by Kuz’menko and Molchanov [15] as early as in 1979 as an example of a manifold where the Liouville property for bounded harmonic functions fails. However, even for this simple manifold, no uniform bounds of the heat kernel was known, except for partial estimates [2], [19]. Our main results - Theorems 1 and 4, provide sharp uniform heat kernel estimates for a rather wide class of connected sums M_{1}#M_{2}#...#M_{k}. In particular, they cover (but are not restricted to) the case when M_{i} are manifolds with non-negative Ricci curvature.

In this section, we will explain our results in the situation when each M_{i} is similar, at infinity, to an Euclidean space. To be precise, let us fix some (large) integer N and define the manifold R^{n} := ℝ^{n}×𝕊^{N−n}, for all integers n from 1 to N. Clearly, R^{n} has topological dimension N whereas, on a large scale, it resembles ℝ^{n}. It is natural to say that the asymptotic dimension of R^{n} is n. It makes sense to consider the connected sums of R^{n}’s with different n since all those manifolds have the same topological dimension N. Hence, the main example of this section is the manifold M = R^{n_{1}}#R^{n_{2}}#...#R^{n_{k}}.

E.B.Davies [9] considered certain one-dimensional model spaces with ends having asymptotic dimensions n_{1}, n_{2}, .... and observed some interesting phenomena for the heat kernel long time behavior. For example, he showed that, if all n_{i} ≠ 2 and some n_{i} > 2 then, for a fixed point x and for t → ∞,

p(t, x, x) ≤ Ct^{−n/2}

where

n = min_{i} {2 + |n_{i} − 2|} .

In particular, the contributions to n of the ends with the asymptotic dimensions 1 and 3 are equal. Indeed, if n_{i} = 1 or 3 then 2 + |n_{i} − 2| = 3. E.B.Davies conjectured that the same is true for connected sums of manifolds, and the positive answer to his conjecture follows from our results (see Corollary 5). Moreover, one can see that there is a certain duality between the manifolds of asymptotic dimensions less than 2 and greater than 2.

For the manifold M = R^{n}, the following estimate of the heat kernel is wellknown:

p(t, x, y) ≍ \frac{1}{min (t^{N/2}, t^{n/2})} exp (−\frac{d^{2}(x, y)}{ct}) ,  (3)

d(x, y) being the geodesic distance on M. Here the sign ≍ means that the ratio of the right-hand side and of the left-hand side is bounded from above and from below by some positive constants, for all values of x, y ∈ M and t > 0. We always follow the convention that the constant c in the Gaussian exponential term may be different for the upper and for the lower bounds.

Consider now the manifold M = R^{n}#R^{m} where both n, m > 2. If t ≤ 1 then the estimate (3) holds again. Assume in the sequel t > 1. For any point x ∈ M, denote

|x| := sup_{z∈K} d(x, z).

The point x ∈ M lies either in the central part K or in one of two ends R^{n} \ K, R^{m} \ K. Let x ∈ R^{n} \ K and y ∈ R^{m} \ K. Then we claim that

p(t, x, y) ≍ (\frac{1}{t^{m/2} |x|^{n−2}} + \frac{1}{t^{n/2} |y|^{m−2}} ) exp (−\frac{d^{2}(x, y)}{ct}) ,  (4)

which is a special case of Theorem 1 below. However, even (4) exhibits some new phenomena. Let us consider two asymptotic regimes.

(a) Long time regime when t → ∞ whereas |x| and |y| stay bounded. Clearly, (4) implies

p(t, x, y) ≍ \frac{1}{t^{min(n,m)/2}}

so that the long time behaviour of the heat kernel is determined by the end with the smallest asymptotic dimension.

(b) Medium time regime when t → ∞ and |x| ≍ |y| ≍ \sqrt{t}. Then (4) implies

p(t, x, y) ≍ \frac{1}{t^{n/2+m/2−1}} .  (5)

In particular, for the manifold M = ℝ^{n}#ℝ^{n}, in the medium time regime, we have

p(t, x, y) ≍ \frac{1}{t^{n−1}} .  (6)

Note that even this estimate is new (see [2] for an upper bound and [19] for a lower bound in the medium time regime). For comparison, if x and y stay in the same end and |x| ≍ |y| ≍ \sqrt{t} then p(t, x, y) ≍ t^{−n/2} as for the heat kernel of ℝ^{n} (see (8) below). Since n − 1 > n/2, the estimate (6) reflects a bottleneck effect: for |x| ≍ |y| ≍ \sqrt{t}, the heat kernel p(t, x, y) is substantially smaller if x, y are in different ends than if they are in the same end.

Each term in (4) has a clear probabilistic meaning. Let X_{t} be the Brownian motion on M, that is, the diffusion generated by the Laplace operator Δ. Then p(t, x, y) is the density of the probability that X_{t} hits y at time t after starting at x. Roughly speaking, there are two possibilities for a continuous path to get from x to y: either X_{t} spends most time in R^{m} or in R^{n}. The probability of the former is of the order \frac{1}{t^{m/2}|x|^{n−2}} and of the latter is of the order \frac{1}{t^{n/2}|y|^{m−2}} .

Another new effect can be observed when considering connected sums of at least three manifolds. Let

M = R^{n_{1}}#R^{n_{2}}#...#R^{n_{k}}

where all n_{i} > 2. Denote

n = min_{i} n_{i} .

Assuming as above that t > 1, x ∈ R^{n_{i}} \ K and y ∈ R^{n_{j}} \ K where i ≠ j, we have the following estimate

p(t, x, y) ≍ (\frac{1}{t^{n_{j}/2} |x|^{n_{i}−2}} + \frac{1}{t^{n_{i}/2} |y|^{n_{j}−2}} + \frac{1}{t^{n/2} |x|^{n_{i}−2} |y|^{n_{j}−2}} ) e^{− \frac{d^{2}(x,y)}{ct}} . (7)

The first two terms here have the same interpretation as above. The third term is dominated by one of the first two if n = n_{i} or n = n_{j} (which is the case, in particular, when k = 2). However, if n < n_{i} and n < n_{j} (which may happen only if k ≥ 3) then the third term dominates, for t large enough. It roughly corresponds to the probability that X_{t} goes from x to y spending a long time in the end with the smallest asymptotic dimension n. Hence, the most probable way of getting from x to y is by looping through the smallest end!

If both points x, y belong to the same end R^{n_{i}} \K, then we have, for all t > 1,

p(t, x, y) ≍ \frac{1}{t^{n/2} |x|^{n_{i}−2} |y|^{n_{i}−2}} e^{− \frac{|x|^{2}+|y|^{2}}{ct}} + \frac{1}{t^{n_{i}/2}} e^{− \frac{d^{2}(x,y)}{ct}} .  (8)

Let us emphasize that in all the estimates above, we have assumed n_{i} > 2 for all i. If n_{i} ≤ 2 for some i then the heat kernel behaves differently as will be discussed in Section 4 - see Theorem 4 and Corollary 5.

The condition n_{i} > 2 is equivalent to the non-parabolicity of R^{n_{i}}. A manifold M is called parabolic if the Brownian motion X_{t} on M is recurrent, and nonparabolic otherwise. The non-parabolicity is equivalent to the existence of a positive fundamental solution of the Laplace operator (see the survey [13] for detailed account of parabolicity).

In our main results - Theorems 1 and 4 below, we will distinguish the cases when all the ends M_{i} are non-parabolic or not.

3 The case when all the ends are non-parabolic

For any manifold M_{i}, denote by d_{i} the geodesic distance on M_{i}, by B_{i}(x, r) the geodesic ball on M_{i} of the radius r centered at x, and by V_{i}(x, r) the Riemannian volume of that ball. Also, let p_{i} be the heat kernel of M_{i}. Note that the nonparabolicity of M_{i} is equivalent to

∫ ^{∞} p_{i}(t, x, x)dt < ∞,  (9)

for some/all x ∈ M_{i}. If M_{i} is geodesically complete and, for some x ∈ M_{i},

∫ ^{∞} \frac{dt}{V_{i}(x, \sqrt{t})} = ∞  (10)

then M_{i} is parabolic - see [5], [10], [14], [20].

Let us introduce the following hypotheses, which in general may be true or not.

(i) The volume doubling condition: for some C > 0 and for all r > 0, x ∈ M_{i},

V_{i}(x, 2r) ≤ CV_{i}(x, r).

(ii) The on-diagonal upper bound of the heat kernel: for some C > 0 and for all t > 0, x ∈ M_{i},

p_{i}(t, x, x) ≤ \frac{C}{V_{i}(x, \sqrt{t})}.  (11)

(iii) The off-diagonal lower bound of the heat kernel: for some C > 0 and for all t > 0, x, y ∈ M,

p_{i}(t, x, y) ≥ \frac{C^{−1}}{V_{i}(x, \sqrt{t})} exp (−C \frac{d_{i}^{2}(x, y)}{t}) .  (12)

For example, all the hypotheses (i), (ii) and (iii) hold if M_{i} has non-negative Ricci curvature - see [16]. Note that, under the hypotheses (i) and (ii), the condition (10) is not only sufficient but also necessary for the parabolicity of M_{i}. Let us also mention that (i) and (ii) imply an off-diagonal upper bound for the heat kernel matching (12) (see [12]) as well as an on-diagonal lower bound matching (11) (see [6]). The necessary and sufficient conditions for (i) and (ii) are given in terms of a relative Faber-Krahn inequality - see [12]. The hypotheses (i),(ii) and (iii) are equivalent to a parabolic Harnack inequality, which, in turn, is equivalent to the volume doubling (i) and a Poincaré inequality - see [17] and [11].

In order to state the heat kernel upper bound on the manifold M, let us introduce the following notation. Denote as above

|x| = sup_{z∈K} d(x, z)

and define the new distances:

1. d_{+}(x, y) is the infimum of lengths of smooth curves connecting x and y in M and intersecting K;

2. d_{∅}(x, y) is the infimum of lengths of smooth curves connecting x and y in M and not intersecting K.

Clearly, both d_{+}(x, y) and d_{∅}(x, y) are larger than or equal to the geodesic distance d(x, y), and one of them is equal to d(x, y). If x and y are at different ends then d_{∅}(x, y) = ∞.

For each i ≥ 1, fix a reference point o_{i} ∈ ∂K ∩ M_{i}, set

W_{i}(r) = V_{i}(o_{i}, r),  (13)
W_{min}(r) = min_{i} W_{i}(r)  (14)

and define the function W (x, r) by

W (x, r) = \{ \begin{matrix}W_{i}(r), & x ∈ E_{i}, \\ W_{min}(r), & x ∈ K. \end{matrix}  (15)

In words, W_{i}(r) is the volume growth function for the manifold M_{i}, W_{min}(r) is the smallest volume growth, and W (x, r) switches between W_{i} and W_{min} depending on the location of x.

Finally, let us define the function H(x, t), for all t > 0 and x ∈ M, by

H(x, t) = \frac{|x|^{2}}{W_{i}(|x|)} + (∫_{|x|^{2}}^{t} \frac{ds}{W_{i}(\sqrt{s})})_{+} , if x ∈ E_{i},  (16)
H(x, t) = 1,  if x ∈ K.  (17)

Theorem 1 Assume that all M_{i}, i = 1, 2, ..., k, are complete non-compact nonparabolic manifolds satisfying (i), (ii) and (iii). Then the heat kernel p(t, x, y) on the connected sum M = M_{1}#M_{2}#....#M_{k} admits the following estimate: for all t > 0 and x, y ∈ M,

p(t, x, y) ≍ (\frac{H(x, t)}{W (y, \sqrt{t})} + \frac{H(y, t)}{W (x, \sqrt{t})} + \frac{H(x, t)H(y, t)}{W_{min}(\sqrt{t})}) e^{− \frac{d^{2}_{+}(x,y)}{ct}} (18)
+\frac{1}{\sqrt{V_{i}(x, \sqrt{t})V_{i}(y, \sqrt{t})}}e^{−^{\frac{d^{2}_{∅}(x,y)}{ct}}} ,  (19)

where the term in (19) refers to the case when both x, y belong to the same end E_{i} (otherwise d_{∅} = ∞).

If all M_{i} satisfy only (i) and (ii) (but not necessarily (iii)) then the upper bound in (18)-(19) is still valid.

In particular, if x ∈ E_{i} and y ∈ E_{j} where i ≠ j then (18)-(19) yield

p(t, x, y) ≍ (\frac{H(x, t)}{W_{j}(\sqrt{t})} + \frac{H(y, t)}{W_{i}(\sqrt{t})} + \frac{H(x, t)H(y, t)}{W_{min}(\sqrt{t})}) e^{− \frac{d^{2}(x,y)}{ct}} ,  (20)

because in this case d_{∅}(x, y) = ∞ and d_{+}(x, y) = d(x, y). If x and y lie at the same end, that is, i = j then we obtain

p(t, x, y) ≍ \frac{H(x, t)H(y, t)}{W_{min}(\sqrt{t})}e^{− \frac{|x|^{2}+|y|^{2}}{ct}} + \frac{1}{V_{i}(x, \sqrt{t})}e^{− \frac{d^{2}(x,y)}{ct}} .  (21)

Let us show examples of computation of the function H.

Example 1 Suppose that, for all i, the volume growth on M_{i} is given by

W_{i}(r) ≍ r^{n_{i}}, for r > 1,

where all n_{i} > 2. The latter implies, in particular, (9) whence M_{i} is non-parabolic. Next, we have

W_{min} ≍ r^{n},

where n = min_{i} n_{i}. Assuming x ∈ E_{i} and |x| > 1, we compute H(x, t) by

H(x, t) ≍ |x|^{2−n_{i}} + (∫_{|x|^{2}}^{t} \frac{ds}{s^{n_{i}/2}} )_{+} ≍ |x|^{2−n_{i}} .

Hence, the estimates (20) and (21) become (7) and (8), respectively.

Example 2 Suppose now that, for all i, the volume growth function is as follows

W_{i}(r) ≍ r^{2} log^{α^{i}} r, r > 2,

where α_{i} > 1 (which ensures the non-parabolicity of M_{i}). Then

W_{min} ≍ r^{2} log^{α} r

where

α = min_{i} α_{i}.

Assuming x ∈ E_{i} and |x| > 2, we compute H(x, t) by

H(x, t) = log^{−α_{i}} |x| + (∫_{|x|^{2}}^{t} \frac{ds}{s log^{α^{i}} s})_{+}

≍ log^{−α_{i}} |x| + (log^{1−α_{i}} |x|^{2} − log^{1−α_{i}} t)_{+} .

If |x| ≥ \sqrt{t} then H(x, t) ≍ log^{−α_{i}} |x|. Assuming y ∈ E_{j} and |x| ≍ |y| ≍ \sqrt{t} (where i ≠ j) we obtain from (20) the following medium time asymptotic

p(t, x, y) ≍ \frac{1}{t log^{α_{i}+α_{j}} t}.

If |x| << \sqrt{t} then H(x, t) ≍ log^{1−α_{1}} |x|. Assuming that |x| , |y| remain bounded while t → ∞, we obtain the long time asymptotic

p(t, x, y) ≍ \frac{1}{t log^{α} t}.

4 The mixed case with both parabolic and nonparabolic ends

We will need here the following additional hypothesis.

(iv) Connectedness of the annuli: for all R large enough, the annulus B_{i}(o_{i}, 4R)\ B_{i}(o_{i}, R/4) is connected^{1}.

So far we have not required that the end E_{i} is connected. Clearly, the connectedness of the annuli is even stronger assumption than the connectedness of E_{i}. Basically, the hypothesis (iv) prohibits the situation when M_{i} consists of two “nearly” disjoint unbounded parts connected only by a rare sequence of small tubes.

It is easy to see that any two-dimensional convex surface in ℝ^{3} satisfies all the hypotheses (i), (ii), (iii) and (iv), provided it is complete and unbounded. Also, all manifolds R^{n} introduced above satisfy these hypotheses.

We start with the following lemma.

1This hypothesis can be slightly relaxed as follows: for any two points x, y both at distance R from o_{i}, there is a continuous path connecting x to y and staying in the annulus B_{i}(o_{i}, 4R) \ B_{i}(o_{i}, R/4).

Lemma 2 Suppose that all M_{i}, i = 1, 2, ..., k are geodesically complete manifolds satisfying (i), (ii), (iii) and (iv). Assume that at least one of them is non-parabolic. Then there exists a positive harmonic function h(x) on M = M_{1}#M_{2}#...#M_{k} such that, for all x ∈ M,

h(x) ≍ 1 + (∫_{1}^{|x|^{2}} \frac{ds}{W (x, \sqrt{s})})_{+}

where W is defined by (13)-(15).

In particular, h(x) ≍ 1 if x stays in a non-parabolic end whereas h(x) → ∞ if x → ∞ within a parabolic end.

For any smooth positive function h(x) on M, one may consider the weighted manifold M̃ = (M, h^{2}) which means that, in addition to the Riemannian structure, M̃ is endowed with the measure μ̃ defined by dμ̃ = h^{2}dμ where μ is the Riemannian measure on M. There is a natural weighted Laplace operator on M̃ defined by

Δ̃ = h^{−2}div(h^{2}∇) ,

which is formally self-adjoint with respect to μ̃. Denote by p̃ the heat kernel of Δ̃. If h(x) is, in addition, harmonic then it is easy to see that

Δ̃ = h^{−1} ∘ Δ ∘ h

which implies

p̃(t, x, y) = \frac{p(t, x, y)}{h(x)h(y)}.  (22)

Let us define the function h_{i}(x) on M_{i} as follows: h_{i}(x) = h(x) if x ∈ E_{i} and h_{i} is extended to K_{i} somehow, to remain smooth and positive. Clearly, we have, for all x ∈ M_{i},

h_{i}(x) ≍ 1 + (∫_{1}^{|x|^{2}} \frac{ds}{W_{i}(\sqrt{s})})_{+} .  (23)

Consider the weighted manifold M̃_{i} = (M_{i}, h_{i}^{2}) and introduce similarly μ̃_{i}, Δ̃_{i} and p̃_{i} (since h_{i} is not necessarily harmonic on M_{i}, the kernel p̃_{i} does not necessarily satisfies the analogue of (22)). Denote Ṽ_{i}(x, r) = μ̃_{i}(B_{i}(x, r)).

Lemma 3 Suppose that all M_{i}, i = 1, 2, ..., k are geodesically complete manifolds satisfying (i), (ii), (iii) and (iv). Assume that at least one of them is non-parabolic. Let h and h_{i} be the functions defined as above. Then each M̃_{i} is non-parabolic and satisfies the conditions (i), (ii) and (iii) (where one should naturally replace V_{i} by Ṽ_{i} and p_{i} by p̃_{i}).

In fact, Theorem 1 remains true for weighted manifolds. Therefore, by Lemma 3, one can apply Theorem 1 to the above setting, estimate p̃, and then recover p by (22). To state the result which we obtain in this way, we need the following notation. Let us set

W̃_{i}(r) := Ṽ_{i}(o_{i}, r)

and define W̃_{min}(r), W̃(x, r) and H̃(x, t) analogously to (14), (15) and (16)-(17) where W_{i} must be replaced everywhere by W̃_{i}.

Theorem 4 Assume that all M_{i}, i = 1, 2, ..., k, are complete non-compact manifolds satisfying (i), (ii), (iii), (iv), and at least one of M_{i}’s is non-parabolic. Then the heat kernel p(t, x, y) on the connected sum M = M_{1}#M_{2}#....#M_{k} admits the following estimate: for all t > 0 and x, y ∈ M,

p(t, x, y) ≍ h(x)h(y) (\frac{H̃(x, t)}{W̃(y, \sqrt{t})} + \frac{H̃(y, t)}{W̃(x, \sqrt{t})} + \frac{H̃(x, t)H̃(y, t)}{W̃_{min}(\sqrt{t})}) e^{− \frac{d^{2}_{+}(x,y)}{ct}}

+\frac{h(x)h(y)}{\sqrt{Ṽ_{i}(x, \sqrt{t})Ṽ_{i}(y, \sqrt{t})}}e^{−^{\frac{d^{2}_{∅}(x,y)}{ct}}} .

Denote

η_{i}(r) := 1 + (∫_{1}^{r^{2}} \frac{ds}{W_{i}(\sqrt{s})})_{+} .  (24)

Then (23) implies h_{i}(x) ≍ η_{i}(|x|). It is possible to prove that

W̃_{i}(r) ≍ η_{i}^{2}(r)W_{i}(r)  (25)

and, if x ∈ E_{i},

H̃(x, t) ≍ \frac{|x|^{2}}{η_{i}^{2}(|x|)W_{i} (|x|)} + \frac{1}{η_{i} (|x|) η_{i} (\sqrt{t})} (∫_{|x|^{2}}^{t} \frac{ds}{W_{i}(\sqrt{s})})_{+} .  (26)

Let us show some examples how to apply Theorem 4 using (24)–(26).

Example 3 Let M_{1} = R^{1} := ℝ_{+} × 𝕊^{N−1}, M_{2} = R^{3} and consider the connected sum M = R^{1}#R^{3}. We have, for r > 1,

W_{1}(r) ≍ r and W_{2}(r) ≍ r^{3}.

By (24), we obtain, for r > 1,

η_{1}(r) ≍ r and η_{2}(r) ≍ 1.

Then we compute W̃_{i} by (25)

W̃_{1}(r) ≍ r^{3} and W̃_{2}(r) ≍ r^{3},

whence

W̃_{min}(r) = min(W̃_{1}(r), W̃_{2}(r)) ≍ r^{3}.

Finally, let us compute H̃(x, t) and H̃(y, t) assuming x ∈ E_{1}, y ∈ E_{2} and |x| , |y| , t > 1. By (26), we obtain

H̃(x, t) ≍ |x|^{−1} and H̃(y, t) ≍ |y|^{−1} .

Hence, Theorem 4 yields, for the chosen x, y and t,

p(t, x, y) ≍ \frac{1}{t^{3/2}} (1 + \frac{|x|}{|y|}) e^{− \frac{d^{2}(x,y)}{ct}} .  (27)

We see that the contributions of both ends R^{1} and R^{3} to the long time behavior of the heat kernel on M are of the same order t^{−3/2}. This may seem surprising, in view of the heat estimate (4) for the manifold R^{n}#R^{m} with n, m > 2, which contains both terms t^{−n/2} and t^{−m/2}. The explanation is that what counts for the manifold R^{1}#R^{3} is the heat kernel long time behavior on R̃^{1} rather than that on R^{1}. However, R̃^{1} has the volume growth W̃_{1}(r) ≍ r^{3} and, therefore, the heat kernel of the order t^{−3/2}. This effect was first observed by E.B.Davies [9] in a model situation of a one-dimensional complex.

Example 4 Let us take M_{1} = R^{2} and M_{2} = R^{3}. We compute as in the previous example, for r > 2 and |x| > 2,

W_{1}(r) ≍ r^{2} and W_{2}(r) ≍ r^{3},
h_{1}(x) ≍ log |x| and h_{2}(x) ≍ 1,
W̃_{1}(r) ≍ r^{2} log^{2} r and W̃_{2}(r) ≍ r^{3}.

If x ∈ E_{1} and y ∈ E_{2} then, for t > 2, we have H̃(y, t) ≍ |y|^{−1} whereas

H̃(x, t) ≍ \frac{1}{log^{2} |x|} + (\frac{1}{2 log |x|} − \frac{1}{log t})_{+} .

Hence, for the chosen x, y, t, we obtain

p(t, x, y) ≍ (\frac{log |x|}{|y| t log^{2} t} + \frac{1}{t^{3/2}} [\frac{1}{log |x|} + (\frac{1}{2} − \frac{log |x|}{log t})_{+}]) e^{− \frac{d^{2}(x,y)}{ct}}

In particular, the long time asymptotic is as follows

p(t, x, y) ≍ \frac{1}{t log^{2} t} ,

whereas the medium time asymptotic (when |x| ≍ |y| ≍ \sqrt{t}) is given by

p(t, x, y) ≍ \frac{1}{t^{3/2} log t} .

In certain situations, the formulas (24), (25) and (26) can be further simplified. Namely, let us assume that, for r large enough,

∫_{1}^{r^{2}} \frac{ds}{W_{i}(\sqrt{s})} ≍ \frac{r^{2}}{W_{i}(r)}.  (28)

Then we have the following relations, for r large enough,

η_{i}(r) ≍ \frac{r^{2}}{W_{i}(r)}
W̃_{i}(r) ≍ \frac{r^{4}}{W_{i}(r)}
H̃(x, t) ≍ |x|^{−2} W_{i}(|x|)

where in the last line x ∈ E_{i} and |x|, t are large enough. For example, (28) holds if W_{i}(r) ≍ r^{α} with 0 < α < 2. For this W_{i}, we obtain

η_{i}(r) ≍ r^{2−α}
W̃_{i}(r) ≍ r^{4−α} = r^{α^{∗}}
H̃(x, t) ≍ |x|^{α−2} = |x|^{2−α^{∗}}

where

α^{∗} := 4 − α.

We see that W̃_{i}(r) and H̃(x, t) look like the corresponding functions on a nonparabolic manifold with volume growth r^{α^{∗}}. Hence, to some extent, the parabolic manifold with volume growth r^{α} can be regarded as dual to the non-parabolic manifold with volume growth r^{α^{∗}}.

Corollary 5 Under the hypotheses of Theorem 4, assume in addition that each manifold M_{i} has volume growth of the order r^{n_{i}} where n_{i} ≠ 2. Denote

n_{i}^{∗} := \{ \begin{matrix}4 − n_{i}, & n_{i} < 2 \\ n_{i}, & n_{i} > 2 \end{matrix}

and

n := min_{1≤i≤k} n_{i}^{∗}.

If x ∈ E_{i}, y ∈ E_{j} (i ≠ j) and |x| , |y| , t are large enough then

p(t, x, y) ≍ (\frac{1}{t^{n_{j}^{∗}/2} |x|^{n_{i}^{∗}−2}} + \frac{1}{t^{n_{i}^{∗}/2} |y|^{n_{j}^{∗}−2}} + \frac{1}{t^{n/2} |x|^{n_{i}^{∗}−2} |y|^{n_{j}^{∗}−2}} )
× |x|^{(2−n_{i})^{+}} |y|^{(2−n_{j})_{+}} e^{− \frac{d^{2}(x,y)}{ct}} .  (29)

In particular, (29) contains the estimate (7) for the manifold R^{n_{1}}#R^{n_{2}}#...#R^{n_{k}} when all n_{i} are larger than 2. The estimate (27) for the manifold R^{1}#R^{3} is also a straightforward consequence of (29).

The long time asymptotic in (29) is determined by the term t^{−n/2}. This was noticed by Davies [9] for an one-dimensional complex, modelling manifolds with ends. If all n_{i} > 2 then n/2 can be interpreted as the exponent of the largest heat kernel of M_{i}’s. However, in general this is not true. It turns out that t^{−n_{i}^{∗}/2} is the rate of decay of the Dirichlet heat kernel of E_{i}, that is, the heat kernel on E_{i} with the vanishing boundary condition on ∂E_{i}. Therefore, the term t^{−n/2} is determined in general by the largest Dirichlet heat kernel on E_{i}’s. In fact, precise estimates of the Dirichlet heat kernel on E_{i} are crucial tools for the proof of the results described above.

Assume n_{i} < 2, n_{j} > 2 and consider the medium time asymptotic regime when |x| ≍ \sqrt{t} and t → ∞, in which case (29) implies

p(t, x, y) ≍ \frac{1}{t^{n_{j}^{∗}/2}} + \frac{1}{t |y|^{n_{j}^{∗}−2}} .  (30)

If also |y| ≍ \sqrt{t} then the first term in (30) is leading. Hence, the medium time asymptotic is determined only by the larger end, in contrast to the case when both ends are non-parabolic (cf. (5)).

However, the most interesting paradoxical effect occurs if we let |y| ≍ 1 in (30). In this case,

p(t, x, y) ≍ t^{−1},  (31)

regardless of the exponents n_{i}, n_{j}! Therefore, if x moves away at the rate \sqrt{t} and y stays near the central part, then the heat kernel p(t, x, y) given by (31) is larger than in the long time regime when both x and y stay near the central part.

The explanation is that if x and y are close to the central part and t is large then the process X_{t} started at x tends to escape to infinity within the larger end so that its chances to loop back to y are relatively small. On the contrary, if X_{t} starts at the point x located at the smaller end at the distance \sqrt{t} from the central part, then it cannot escape to infinity within this end because of its parabolicity. Hence, it moves towards the central part and hits y in time t with a higher probability.

The hypothesis of Theorem 4 and Corollary 5 that at least one end M_{i} is non-parabolic is equivalent to the non-parabolicity of M - see [10]. If all M_{i}’s are parabolic and, thus, M is also parabolic, then the estimate (29) cannot be true. Indeed, the long time decay t^{−n/2} given by (29) implies (9), which contradicts the parabolicity of M.

The case when all M_{i} are parabolic will be addressed elsewhere.

Acknowledgment Both authors were supported by the travel grants of the EU Stochastic Analysis Network.

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