Ruifeng Qiu

The Heegaard distances cover all nonnegative integers

In this paper, we prove that (1) For any integers

HEEGAARD GENERA OF ANNULAR 3-MANIFOLDS

n\geq 1

Unstabilized self-amalgamation of a Heegaard splitting

and

The self-amalgamation of high distance Heegaard splittings is always efficient

g\geq 2

The Heegaard genera of surface sums

, there is a closed 3-manifold

A note on the uniqueness of unstabilized Heegaard splittings of amalgamated 3-manifolds

M_{g}^{n}

The ∂-stabilization of a Heegaard splitting with distance at least 6 is unstabilized

which admits a distance

A note on tunnel number of composite knots

n

Additivity of Heegaard genera of bounded surface sums

Heegaard splitting of genus

The subset of R3 realizing metrics on the curve complex

g