Video: In order to harden the edge of steel, swords are often heated until they're impossibly hot and then quenched in liquid to rapidly cool down. This creates a much stronger grain structure in the steel which obviously leads to a much stronger sword. An interesting thing that happens during this quenching process is how the sword dramatically bends before it snaps back into shape (with a slight upward tilt). Watch it in the video below. You can see the katana get dipped into the liquid, curve downward in shock and then immediately lift itself back up as if were alive in a matter of seconds.
The reason this happens is because of the two different sides of the sword. A katana has a single edge blade which means that one side (the edge) is going to be thinner while the other side (the back) is going to be thicker. The thinner side gets cooled first, which results in the curve downward as it contracts. As the thicker back side gets cooled, it curves back upward to give the katana its iconic shape.